Find the limit.
-$\lim _ { x \rightarrow \theta ^ { + } } x \tan \left( \frac { 3 \pi } { 2 } + x \right)$
A) $- 1$
B) $- \frac { 1 } { 2 }$
C) 1
D) 0

Question

Quizplus · Accepted Answer

The given limit is $\lim _ { x \rightarrow \theta ^ { + } } x \tan \left( \frac { 3 \pi } { 2 } + x \right)$. As $x$ approaches $0$ from the positive side, $\tan \left( \frac { 3 \pi } { 2 } + x \right)$ approaches $\tan \left( \frac { 3 \pi } { 2 } \right)$, which is undefined because $\tan$ function has a vertical asymptote at $\frac { 3 \pi } { 2 }$. However, near $\frac { 3 \pi } { 2 }$, $\tan(x)$ behaves like $\frac{1}{x-\frac{3\pi}{2}}$, which approaches negative infinity as $x$ approaches $\frac{3\pi}{2}$ from the right. Thus, the expression behaves like $x \cdot (-\infty)$ as $x$ approaches $0$ from the positive side, which tends to $-0$, effectively being $-1$ when considering the behavior of $\tan$ near its vertical asymptote.

Find the Limit lim⁡x→θ+xtan⁡(3π2+x)\lim _ { x \rightarrow \theta ^ { + } } x \tan \left( \frac { 3 \pi } { 2 } + x \right)limx→θ+​xtan(23π​+x)

Find the Limit $\lim _ { x \rightarrow \theta ^ { + } } x \tan \left( \frac { 3 \pi } { 2 } + x \right)$