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Dimas [21]
3 years ago
7

Limit question: lim x-->pi ((e^sinx)-1)/(x-pi)

Mathematics
1 answer:
aleksandr82 [10.1K]3 years ago
4 0
\displaystyle\lim_{x\to\pi}\dfrac{e^{\sin x}-1}{x-\pi}

Notice that if f(x)=e^{\sin x}, then f(\pi)=e^{\sin\pi}=e^0=1. Recall the definition of the derivative of a function f(x) at a point x=c:

f'(c):=\displaystyle\lim_{x\to c}\frac{f(x)-f(c)}{x-c}

So the value of this limit is exactly the value of the derivative of f(x)=e^{\sin x} at x=\pi.

You have

f'(x)=\cos x\,e^{\sin x}\implies f'(\pi)=\cos\pi\,e^{\sin\pi}=-1
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