Q. 32 - Calculus

Question

Consider the graph  of the function:

Find the limits of function.

Step-by-Step Solution

Answer

The limits of the function are -1

Step 1. Given graph

Step 2. Finding limits to the left

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

Step 3. Finding limits to the right

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

Step 4. Finding limits to the Point

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