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JEE Maths Limit, Continuity and Differentiability

Limit, Continuity and Differentiability PDF Notes, Important Questions and Synopsis

 

SYNOPSIS

  1. The expected value of the function as dictated by the points to the left of a given point defines the left-hand limit of the function at that point. The limitbegin mathsize 12px style limit as straight x rightwards arrow straight a to the power of minus of straight f left parenthesis straight x right parenthesis end style is the expected value of ‘f’ at x = a, given the values of ‘f’ near ‘x’ to the left of ‘a’.
  2. The expected value of the function as dictated by the points to the right of a given point defines the right-hand limit of the function at that point. The limitbegin mathsize 12px style limit as straight x rightwards arrow straight a to the power of plus of straight f left parenthesis straight x right parenthesis end style is the expected value of ‘f’ 
    at x = a, given the values of ‘f’ near ‘x’ to the right of ‘a’.

  3. Let y = f(x) be a function. Suppose that ‘a’ and ‘L’ are numbers such that as ‘x’ gets closer and closer to ‘a’, f(x) gets closer and closer to ‘L’. We say that the limit of f(x) at x = a is L, 
    i.e.begin mathsize 12px style limit as straight x rightwards arrow straight a of straight f left parenthesis straight x right parenthesis equals straight L end style

  4. Limit of a function at a point is the common value of the left- and right-hand limit if they coincide,
    i.e. if begin mathsize 12px style limit as straight x rightwards arrow straight a to the power of minus of straight f left parenthesis straight x right parenthesis end style=  begin mathsize 12px style limit as straight x rightwards arrow straight a to the power of plus of straight f left parenthesis straight x right parenthesis end style.

  5. Real-life examples of LHL and RHL:
    i. If a car starts from rest and accelerates to 60 km/hr in 8 seconds, which means the initial speed of the car is 0 and it reaches 60 km/hr 8 seconds after the start. On recording the speed of the car, we can see that this sequence of numbers is approaching 60 km in such a way that each member of the sequence is less than 60. This sequence illustrates the concept of approaching a number from the left of that number.

    ii. Boiled milk which is at a temperature of 100 degrees is placed on a shelf. The temperature goes on dropping till it reaches room temperature. As the time duration increases, temperature of milk t approaches room temperature, say 30°C .This sequence illustrates the concept of approaching a number from the right of that number.

  6. Let f and g be two functions such that both Error converting from MathML to accessible text.and begin mathsize 12px style limit as straight x rightwards arrow straight a of text    end text straight g left parenthesis straight x right parenthesis end styleexist, then
    i. Limit of the sum of two functions is the sum of the limits of the functions, 
       i.e. begin mathsize 12px style limit as straight x rightwards arrow straight a of text    end text left square bracket straight f left parenthesis straight x right parenthesis plus straight g left parenthesis straight x right parenthesis right square bracket equals limit as straight x rightwards arrow straight a of text    end text straight f left parenthesis straight x right parenthesis plus limit as straight x rightwards arrow straight a of text    end text straight g left parenthesis straight x right parenthesis. end style
    ii. Limit of the difference of two functions is the difference of the limits of the functions, 
        i.e. begin mathsize 12px style limit as straight x rightwards arrow straight a of text    end text left square bracket straight f left parenthesis straight x right parenthesis minus straight g left parenthesis straight x right parenthesis right square bracket equals limit as straight x rightwards arrow straight a of text    end text straight f left parenthesis straight x right parenthesis minus limit as straight x rightwards arrow straight a of text    end text straight g left parenthesis straight x right parenthesis. end style
    iii. Limit of the product of two functions is the product of the limits of the functions, 
        i.e. begin mathsize 12px style limit as straight x rightwards arrow straight a of text    end text left square bracket straight f left parenthesis straight x right parenthesis. straight g left parenthesis straight x right parenthesis right square bracket equals limit as straight x rightwards arrow straight a of text    end text straight f left parenthesis straight x right parenthesis. text   end text limit as straight x rightwards arrow straight a of text    end text straight g left parenthesis straight x right parenthesis. end style
    iv. Limit of the quotient of two functions is the quotient of the limits of the functions 
        (whenever the denominator is non-zero), 
          begin mathsize 12px style straight i. straight e. text    end text limit as straight x rightwards arrow straight a of fraction numerator straight f left parenthesis straight x right parenthesis over denominator straight g left parenthesis straight x right parenthesis end fraction equals fraction numerator limit as straight x rightwards arrow straight a of text    end text straight f left parenthesis straight x right parenthesis over denominator limit as straight x rightwards arrow straight a of text    end text straight g left parenthesis straight x right parenthesis end fraction end style

  7. Error converting from MathML to accessible text.
    begin mathsize 12px style limit as straight x rightwards arrow straight a of open vertical bar 1 plus straight f open parentheses straight x close parentheses close vertical bar to the power of fraction numerator 1 over denominator straight g open parentheses straight x close parentheses end fraction end exponent equals straight e to the power of limit as straight x rightwards arrow straight a of fraction numerator straight f open parentheses straight x close parentheses over denominator straight g open parentheses straight x close parentheses end fraction end exponent end style

  8. begin mathsize 12px style If space limit as straight x rightwards arrow straight a of straight f open parentheses straight x close parentheses equals 1 space and space limit as straight x rightwards arrow straight a of straight g open parentheses straight x close parentheses equals ∞  such space that end style
    begin mathsize 12px style table attributes columnalign left end attributes row cell limit as straight x rightwards arrow straight a of open vertical bar straight f open parentheses straight x close parentheses minus 1 close vertical bar straight g open parentheses straight x close parentheses text  exists, then, end text end cell row cell limit as straight x rightwards arrow straight a of straight f open parentheses straight x close parentheses to the power of straight g open parentheses straight x close parentheses end exponent equals straight e to the power of limit as straight x rightwards arrow straight a of open vertical bar straight f open parentheses straight x close parentheses minus 1 close vertical bar straight g open parentheses straight x close parentheses end exponent end cell end table end style

  9. For any positive integer n,
    begin mathsize 12px style limit as straight x rightwards arrow straight a of text    end text fraction numerator straight x to the power of straight n minus straight a to the power of straight n over denominator straight x minus straight a end fraction equals na to the power of straight n minus 1 end exponent end style

  10. Limit of a polynomial function can be computed using substitution or algebra of limits.

  11. Methods used to evaluate algebraic limits:
    i. Direct substitution 
    ii. Factorisation 
    iii. Rationalisation
    iv. Using some standard limits
    v. Evaluation of algebraic limits at infinity

  12. Let ‘f’ and ‘g’ be two real-valued functions with the same domain such that f(x)  g(x) for all x in the domain of definition. For some ‘a’, if both  begin mathsize 12px style limit as straight x rightwards arrow straight a of text    end text straight f left parenthesis straight x right parenthesis end style and  begin mathsize 12px style limit as straight x rightwards arrow straight a of text    end text straight g left parenthesis straight x right parenthesis end styleexist, then  begin mathsize 12px style limit as straight x rightwards arrow straight a of text    end text straight f left parenthesis straight x right parenthesis less or equal than limit as straight x rightwards arrow straight a of text    end text straight g left parenthesis straight x right parenthesis end style.

  13. Let f, g and h be real functions such that f(x)  g(x)  h(x) for all x in the common domain of definition. For some real number a, ifbegin mathsize 12px style limit as straight x rightwards arrow straight a of end style   f(x) = ℓ=  begin mathsize 12px style limit as straight x rightwards arrow straight a of end style h(x), thenbegin mathsize 12px style limit as straight x rightwards arrow straight a of end style   g(x) = ℓ. 

  14. 14. If ‘f’ is a real-valued function and ‘a’ is a point in its domain of definition, then the derivative of ‘f’ at ‘a’ is defined by

    begin mathsize 12px style limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction end style
    Provided this limit exists and is finite. Derivative of f(x) at ‘a’ is denoted by f’(a).

  15. A function is differentiable in its domain if it is always possible to draw a unique tangent at every point on the curve.

  16. A function f(x) is said to be continuous at a point c if
    begin mathsize 12px style limit as straight x rightwards arrow straight c to the power of minus of straight f left parenthesis straight x right parenthesis equals limit as straight x rightwards arrow straight c to the power of plus of straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis end style

  17. A real function f is said to be continuous if it is continuous at every point in the domain of f.

  18. If f and g are real-valued functions such that (f o g) is defined at c, then
    begin mathsize 12px style open parentheses straight f text   end text straight o text   end text straight g close parentheses open parentheses straight x close parentheses equals straight f open parentheses straight g open parentheses straight x close parentheses close parentheses. end style
    If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c.

  19. A function f is differentiable at a point c if left-hand derivative (LHD) = right-hand derivative (RHD),

    begin mathsize 12px style limit as straight h rightwards arrow 0 to the power of minus of fraction numerator straight f left parenthesis straight c plus straight h right parenthesis minus straight f left parenthesis straight c right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of plus of fraction numerator straight f left parenthesis straight c plus straight h right parenthesis minus straight f left parenthesis straight c right parenthesis over denominator straight h end fraction end style

  20. If a function f is differentiable at every point in its domain, then

    begin mathsize 12px style limit as straight h rightwards arrow 0 of fraction numerator straight f open parentheses straight x plus straight h close parentheses minus straight f open parentheses straight c close parentheses over denominator straight h end fraction space or space limit as straight h rightwards arrow 0 of fraction numerator straight f open parentheses straight x minus straight h close parentheses minus straight f open parentheses straight c close parentheses over denominator negative straight h end fraction end styleis called the derivative or differentiation of f at x and is 
    denoted bybegin mathsize 12px style straight f to the power of apostrophe open parentheses straight x close parentheses text  or  end text fraction numerator text d end text over denominator text dx end text end fraction straight f open parentheses straight x close parentheses end style  .

  21. If begin mathsize 12px style LHD not equal to RH text   end text straight D end style , then the function f(x) is not differentiable at x = c.

  22. Geometrical meaning of differentiability:

    The function f(x) is differentiable at a point P if there exists a unique tangent at point P. In other words, f(x) is differentiable at a point P if the curve does not have P as its corner point.

  23. A function is said to be differentiable in an interval (a, b) if it is differentiable at every point of (a, b).

  24. A function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b].

  25. Chain rule of differentiation: If f is a composite function of two functions u and v such that f = v(t) and t = u(x), also both  begin mathsize 12px style dv over dt space and space dt over dx end style exist, then begin mathsize 12px style dv over dx equals dv over dt. dt over dx end style .

  26. Logarithm of a to the base b is x, i.e. logba = x if bx = a, where b > 1 is a real number. 

  27. Functions of the form x = f(t) and y = g(t) are parametric functions.

  28. Rolle’s theorem: If f : [a, b]  R is continuous on [a, b] and differentiable on (a, b) such that f(a) = 

    f(b), then there exists some c in (a, b) such that f’(c) = 0.

  29. Mean-value theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b), then there 
    exists some c in (a, b) such thatbegin mathsize 12px style straight f apostrophe left parenthesis straight c right parenthesis equals limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight b right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight b minus straight a end fraction end style  .

  30. A function is continuous at x = c if the function is defined at x = c and the value of the function at x = c equals the limit of the function at x = c. 

  31. If function f is not continuous at c, then f is discontinuous at c and c is called the point of discontinuity of f.

  32. Every polynomial function is continuous.

  33. The greatest integer function [x] is not continuous at the integral values of x.

  34. Every rational function is continuous.

  35. Algebra of continuous functions
    i. Let f and g be two real functions continuous at a real number c, then
    f + g is continuous at x = c.
    ii. f - g is continuous at x = c.
    iii. f . g is continuous at x = c.
    iv. begin mathsize 12px style open parentheses straight f over straight g close parentheses end styleis continuous at x = c, [provided g(c)  0]
    v. kf is continuous at x = c, where k is a constant.

  36. Consider the following functions:
    i. Constant function
    ii. Identity function
    iii. Polynomial function
    iv. Modulus function
    v. Exponential function
    vi. Sine and cosine functions
    These functions are continuous everywhere.

  37. Consider the following functions:
    i. Logarithmic function
    ii. Rational function
    iii. Tangent, cotangent, secant and cosecant functions
    These functions are continuous in their domains.

  38. If f is a continuous function, then  begin mathsize 12px style open vertical bar straight f close vertical bar space and space straight 1 over straight f end style are continuous in their domains.

  39. Inverse functions   begin mathsize 12px style sin to the power of negative 1 end exponent straight x comma cos to the power of negative 1 end exponent straight x comma tan to the power of negative 1 end exponent straight x comma cot to the power of negative 1 end exponent straight x comma cosec to the power of negative 1 end exponent straight x text  and sec end text to the power of text -1 end text end exponent straight x end styleare continuous functions in their respective domains.

  40. The derivative of a function f with respect to x is f’(x) which is given bybegin mathsize 12px style straight f to the power of apostrophe left parenthesis straight x right parenthesis equals limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction end style  .

  41. If a function f is differentiable at a point c, then it is also continuous at that point.

  42. Every differentiable function is continuous, but the converse is not true.

  43. Every polynomial function is differentiable at each begin mathsize 12px style straight x element of straight R end style .

  44. Every constant function is differentiable at each begin mathsize 12px style straight x element of straight R end style .

  45. The chain rule is used to differentiate composites of functions.

  46. The derivative of an even function is an odd function and that of an odd function is an even function.

  47. Algebra of derivatives:
    If u and v are two functions which are differentiable, then
    i. begin mathsize 12px style left parenthesis straight u plus-or-minus straight v right parenthesis apostrophe equals straight u apostrophe plus-or-minus straight v apostrophe text     end text left parenthesis Sum text    end text and text    end text Difference text   end text Formula right parenthesis end style   
    ii. begin mathsize 12px style left parenthesis uv right parenthesis apostrophe equals straight u apostrophe straight v plus uv apostrophe text        end text left parenthesis text Leibnitz rule or  end text Product text    end text rule right parenthesis end style
    iii. begin mathsize 12px style open parentheses straight u over straight v close parentheses to the power of apostrophe equals fraction numerator straight u apostrophe straight v minus uv apostrophe text   end text over denominator straight v squared end fraction text    end text comma text  v end text not equal to text 0, end text left parenthesis Quotient text    end text rule right parenthesis end style

  48. Implicit functions:
    If it is not possible to separate the variables x and y, then the function f is known as an implicit function.

  49. Exponential function: A function of the form y = f(x) = bx, where base b > 1.
    i. Domain of the exponential function is R, the set of all real numbers.
    ii. The point (0, 1) is always on the graph of the exponential function.
    iii. The exponential function is ever increasing.

  50. The exponential function is differentiable at each begin mathsize 12px style straight x element of straight R end style .

  51. Properties of logarithmic functions:
    i. Domain of log function is R+.
    ii. The log function is ever increasing.
    iii. For ‘x’ very near to zero, the value of log x can be made lesser than any given real number.

  52. Logarithmic differentiation is a powerful technique to differentiate functions of the form f(x) = [u(x)]v(x). Here, both f(x) and u(x) need to be positive.

  53. To find the derivative of a product of a number of functions or a quotient of a number of functions, take the logarithm of both sides first and then differentiate.

  54. Logarithmic differentiation:
    y = ax
    Taking logarithm on both sides,
    begin mathsize 12px style logy equals loga to the power of straight x end style
    Using the property of logarithms,
    begin mathsize 12px style logy equals xloga end style
    Now differentiating the implicit function,
    begin mathsize 12px style 1 over straight y. dy over dx equals loga end style
    begin mathsize 12px style dy over dx equals yloga equals straight a to the power of straight x loga end style

  55. The logarithmic function is differentiable at each point in its domain.

  56. Trigonometric and inverse-trigonometric functions are differentiable in their respective domains.

  57. The sum, difference, product and quotient of two differentiable functions are differentiable.

  58. The composition of a differentiable function is a differentiable function. 

  59. A relation between variables x and y expressed in the form x = f(t) and y = g(t) is the parametric form with t as the parameter. Parametric equation of parabola y2 = 4ax is x = at2, y = 2at.

  60. Differentiation of an infinite series: If f(x) is a function of an infinite series, then to differentiate the function f(x), use the fact that an infinite series remains unaltered even after the deletion of a term.

  61. Parametric differentiation:
    Differentiation of the functions of the form begin mathsize 12px style straight x end style= f(t) and y = g(t):
    begin mathsize 12px style table attributes columnalign left end attributes row cell dy over dx equals fraction numerator dy over dt over denominator dx over dt end fraction end cell row cell dy over dx equals dy over dt cross times dt over dx end cell end table end style

  62. Let u = f(x) and v = g(x) be two functions of x. Hence, to find the derivative of f(x) with respect g(x), we use the following formula:
    begin mathsize 12px style du over dv equals fraction numerator du over dx over denominator dv over dx end fraction end style

  63. If y = f(x) and begin mathsize 12px style dy over dx end style  = f’(x) and if f’(x) is differentiable, then   

    begin mathsize 12px style straight d over dx open parentheses dy over dx close parentheses equals fraction numerator straight d squared straight y over denominator dx squared end fraction end style or f’’(x) is the second-order derivative of y with respect to x.

  64. If x = f(t) and y = g(t), then

    begin mathsize 12px style table attributes columnalign left end attributes row cell fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight d over dx open curly brackets fraction numerator straight g to the power of apostrophe open parentheses straight t close parentheses over denominator straight f to the power of apostrophe open parentheses straight t close parentheses end fraction close curly brackets end cell row cell or text    end text fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight d over dt open curly brackets fraction numerator straight g to the power of apostrophe open parentheses straight t close parentheses over denominator straight f to the power of apostrophe open parentheses straight t close parentheses end fraction close curly brackets times dt over dx text end text end cell row cell text or,  end text fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator straight f to the power of apostrophe open parentheses straight t close parentheses straight g to the power of double apostrophe open parentheses straight t close parentheses minus straight g to the power of apostrophe open parentheses straight t close parentheses straight f to the power of double apostrophe open parentheses straight t close parentheses over denominator open curly brackets straight f to the power of apostrophe open parentheses straight t close parentheses close curly brackets cubed end fraction end cell end table end style