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

evaluate the following limit. if the limit is infinite, enter infinity or -infinity. if the limit does not exist, enter dne. lim_{x→0+} csc(x)

evaluate the following limit. if the limit is infinite, enter infinity or -infinity. if the limit does not exist, enter dne. lim_{x→0+} csc(x)

Answer

Explanation:

Step1: Recall csc(x) definition

$\csc(x)=\frac{1}{\sin(x)}$, so we need to find $\lim_{x\rightarrow0^{+}}\frac{1}{\sin(x)}$.

Step2: Analyze sin(x) behavior

As $x\rightarrow0^{+}$, $\sin(x)\rightarrow0^{+}$ (since $\sin(x)$ is positive for small positive $x$ - values and $\sin(0) = 0$).

Step3: Evaluate the limit

Let $y = \sin(x)$. Then we have $\lim_{y\rightarrow0^{+}}\frac{1}{y}$. As $y$ approaches $0$ from the positive - side, $\frac{1}{y}$ approaches infinity.

Answer:

infinity