Question

Graph each function. Use the graph to find the indicated limit, if it exists.

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }f\left(x\right),f\left(x\right)=\sin \left(x\right)$

Short answer

The limit is$1$.

Step-by-step solution

Step 1. Given Information  

We are given a function $f\left(x\right)=\sin \left(x\right)$ and we need to graph the function and find $\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }f\left(x\right)$.

Step 2. Plotting the graph 

So, the graph of the function $f\left(x\right)=\sin \left(x\right)$is

Step 3. Finding the limit   

To find the value of $\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }f\left(x\right)$we need to find the value of the function at $x=\frac{\mathrm{\pi }}{2}$.

$$\begin{gathered} f\left(x\right)=\sin \left(x\right) \\ \Rightarrow f\left(\frac{\mathrm{\pi }}{2}\right)=\sin \left(\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow f\left(\frac{\mathrm{\pi }}{2}\right)=1 \\ \therefore \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }f\left(x\right)=1 \end{gathered}$$