Off the limit

Calculus Level 3

$\large \lim_{x\rightarrow 0} \frac {\ln(\tan(\pi/4 +2x))}{\sin(x)} = \ ?$

0 1 Undefined 6 Does not exist 4 3 2

1 solution

Chew-Seong Cheong
Dec 20, 2016

$\begin{aligned} L & = \lim_{x \to 0} \frac {\ln \left(\tan \left(\frac \pi 4 + 2x \right) \right)}{\sin x} & \small \color{#3D99F6} \text{A 0/0 cases, L'Hôpital's rule applies.} \\ & = \lim_{x \to 0} \frac {\frac {2\sec^2 \left(\frac \pi 4 + 2x \right)}{\tan \left(\frac \pi 4 + 2x \right)}}{\cos x} & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }x \\ & = \frac {\frac {2(\sqrt 2)^2}1}1 = \boxed{4} \end{aligned}$

Off the limit

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