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Calculus Level 4

$\large \displaystyle\lim_{x\to 0^+} \ln((\sin x)^{\tan x}) = \, ?$

None of these choices -1 The limit does not exist 2 1 0 -2

1 solution

Tom Engelsman
Dec 26, 2016

Let us write $ln sin(x)^{tan(x)}$ as $tan(x) \cdot ln(sin(x)) = \frac{ln(sin(x))}{cot(x)}.$ Applying L'Hopital's Rule twice results in:

$\frac{ln(sin(x))}{cot(x)} \Rightarrow \frac{cot(x)}{-csc^2(x)} \Rightarrow \frac{-csc^2(x)}{(-2csc(x))(-csc(x)cot(x))} = -\frac{1}{2cot(x)}.$

which the limit approaches zero as $x \rightarrow 0.$