Question
Tue August 28, 2012 By: Abhijith C
 

Find K so that the function is continuous at x=0 f(x)={sinx+cosx/x , x=0 {k , x=0

Expert Reply
Wed August 29, 2012
Let f(x) = {(sin x + cos x) / x, if x is not equal to 0
               {k, if x = 0

f is continuous at x = 0.

Lim (x->0) f(x) = f(0)

f(0) = k

Lim (x->0) [(sin x + cos x) / x]
Lim (x->0) [(sin x / x) + (cos x / x)]
Lim (x->0) (sin x / x) + Lim (x->0) (cos x / x)
1 + Lim (x->0) (cos x / x)

Note that Lim (x->0) (cos x / x) = (1/0) which is infinity

Thus, the lim (x->0) (sin x + cos x) / x does not exist. Thus, there is no value of k that would make f(x) continuous at x = 0.
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