Step-by-step Solution

Evaluate the limit of $\frac{\ln\left(1+\sin\left(x\right)\right)}{\tan\left(x\right)}$ as $x$ approaches $\pi $

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Step-by-step explanation

Problem to solve:

$\lim_{x\to\pi}\left(\frac{\ln\left(1+\sin\left(x\right)\right)}{\tan\left(x\right)}\right)$

Learn how to solve limits problems step by step online.

$\lim_{x\to\pi }\left(\frac{\frac{d}{dx}\left(\ln\left(1+\sin\left(x\right)\right)\right)}{\frac{d}{dx}\left(\tan\left(x\right)\right)}\right)$

Unlock this full step-by-step solution!

Learn how to solve limits problems step by step online. Evaluate the limit of (ln(1+sin(x))/(tan(x) as x approaches pi. If we try to evaluate the limit directly, it results in indeterminate form. Then we need to apply L'Hôpital's rule. The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if {f(x) = tan(x)}, then {f'(x) = sec^2(x)\cdot D_x(x)}. The derivative of the linear function is equal to 1. The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}.

Final Answer

$-1$
$\lim_{x\to\pi}\left(\frac{\ln\left(1+\sin\left(x\right)\right)}{\tan\left(x\right)}\right)$

Main topic:

Limits

Related formulas:

6. See formulas

Steps:

9

Time to solve it:

~ 0.03 s (SnapXam)