We have an answer to "Use continuity to evaluate the limit. limx→πsin⁡(x+sin⁡x)"

avissidep

Answered question

2021-10-17

Use continuity to evaluate the limit.

lim_{x \to π} \sin (x + \sin x)

au4gsf

Skilled2021-10-18Added 95 answers

x and

\sin x

are continuous, therefore

x + \sin x

is continuous
Recall that : f(g(x)) is continuous if f(x) and g(x) are continuous
Therefore

\sin (x + \sin x)

is a continuous function
Recall that : If f(x) is continuous at x=a, then

lim_{x \to a} f (x) = f (a)

Therefore

lim_{x \to π} \sin (x + \sin x) = \sin (π + \sin π) = \sin (π + 0) = \sin π = 0

Result:

lim_{x \to π} \sin (x + \sin x) = 0

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