evaluate the following limit. if the limit is infinite, enter infinity or -infinity. if the limit does not…

Explanation:

Step1: Recall the definition of the tangent function

The tangent function is defined as $\tan(x)=\frac{\sin(x)}{\cos(x)}$. We want to find $\lim_{x\rightarrow\frac{3\pi}{2}^+}\tan(x)=\lim_{x\rightarrow\frac{3\pi}{2}^+}\frac{\sin(x)}{\cos(x)}$.

Step2: Analyze the behavior of sine and cosine near $x = \frac{3\pi}{2}$

As $x\rightarrow\frac{3\pi}{2}^+$, we know that $\sin(x)\rightarrow - 1$ and $\cos(x)\rightarrow0^-$.

Step3: Evaluate the limit

When we consider the quotient $\frac{\sin(x)}{\cos(x)}$ with $\sin(x)\rightarrow - 1$ and $\cos(x)\rightarrow0^-$, we have $\lim_{x\rightarrow\frac{3\pi}{2}^+}\frac{\sin(x)}{\cos(x)}=\infty$.

Answer:

infinity