Q. 34 - Calculus

Question

Consider the graph  of the function:

Find the limits of function.

Step-by-Step Solution

Answer

The limit of the function is 2.

Step 1. Given Graph

Step 2. Finding limits to the left

$$\begin{gathered} \mathrm{Here} \mathrm{the} \mathrm{limit} \mathrm{expression} \mathrm{is}: \\ \underset{x->0^{-}}{\lim } f\left(x\right) \\ \mathrm{From} \mathrm{the} \mathrm{graph}, \mathrm{the} \mathrm{functional} \mathrm{value} \mathrm{to} \mathrm{the} \mathrm{left} \mathrm{of} x=0 Is f\left(x\right)=2 \\ \mathrm{Therefore}, \underset{x->0^{-}}{\lim } f\left(x\right) =2. \end{gathered}$$

Step 3. Finding limits to the right

$$\begin{gathered} \mathrm{Here} \mathrm{the} \mathrm{limit} \mathrm{expression} \mathrm{is}: \\ \underset{x->1}{\lim } f\left(x\right) \\ \mathrm{From} \mathrm{the} \mathrm{graph}, \mathrm{the} \mathrm{functional} \mathrm{value} \mathrm{to} \mathrm{the} \mathrm{right} \mathrm{of} \mathrm{x}=1 \mathrm{Is} \mathrm{f}\left(\mathrm{x}\right)=1 \\ \mathrm{Therefore}, \underset{x->1}{\lim } f\left(x\right) =1. \end{gathered}$$

Step 4. Finding limits to the points

$$\begin{gathered} \mathrm{Here} \mathrm{the} \mathrm{limit} \mathrm{expression} \mathrm{is}: \\ \underset{x->\infty }{\lim } f\left(x\right) \\ \mathrm{From} \mathrm{the} \mathrm{graph}, \mathrm{the} \mathrm{functional} \mathrm{valueof} x=\infty is f\left(x\right)=1 \\ \mathrm{Therefore}, \underset{x->\infty }{\lim } f\left(x\right) =1. \\ \mathrm{Hence} \mathrm{limit} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{is} 2 \end{gathered}$$