Question

use lhôpitals rule to find the following limit.
lim (x + π)csc x
x→ - π⁻

lim (x + π)csc x=​(type an exact answer.)
x→ - π⁻

Explanation:

Step1: Rewrite the function

Recall that $\csc x=\frac{1}{\sin x}$, so $(x + \pi)\csc x=\frac{x+\pi}{\sin x}$. As $x\to-\pi^{-}$, we have the indeterminate - form $\frac{0}{0}$.

Step2: Apply L'Hopital's Rule

Differentiate the numerator and the denominator. The derivative of $u = x+\pi$ with respect to $x$ is $u^\prime=1$, and the derivative of $v=\sin x$ with respect to $x$ is $v^\prime=\cos x$.
By L'Hopital's Rule, $\lim_{x\to-\pi^{-}}\frac{x + \pi}{\sin x}=\lim_{x\to-\pi^{-}}\frac{1}{\cos x}$.

Step3: Evaluate the limit

Substitute $x =-\pi$ into $\frac{1}{\cos x}$. We know that $\cos(-\pi)=-1$. So $\lim_{x\to-\pi^{-}}\frac{1}{\cos x}=-1$.

Answer:

$-1$