Q. 33 - 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

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

Step 3. Finding limits to the right

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

Step 4.Finding limits to the points

$$\begin{gathered} \mathrm{Here} \mathrm{the} \mathrm{limit} \mathrm{expression} \mathrm{is}: \\ \underset{x->2^{-}}{\lim } f\left(x\right) \\ \mathrm{From} \mathrm{the} \mathrm{graph}, \mathrm{the}re are two \mathrm{functional} \mathrm{at} \mathrm{the} \mathrm{point} \mathrm{x}=2 \mathrm{but} \mathrm{they} \mathrm{are} \mathrm{for} \mathrm{left} \mathrm{or} \mathrm{right} \mathrm{limit}s \\ \mathrm{There} \mathrm{is} \mathrm{no} \mathrm{functional} \mathrm{value} at x=2 \\ \mathrm{Therefore}, \underset{x->2^{-}}{\lim } f\left(x\right) =\mathrm{No} \mathrm{value} \\ \mathrm{Hence} \mathrm{the} \mathrm{limit} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{is} \mathrm{f}\left(2\right)=2. \end{gathered}$$