Chapter 2.1, Problem 11E

To determine

To find: The value of $\underset{y\to -3}{\lim }\left(\frac{y^2+4y+3}{y^2-3}\right)$ with the help of substitution. Check the results graphically.

Answer to Problem 11E

The value of $\underset{y\to -3}{\lim }\left(\frac{y^2+4y+3}{y^2-3}\right)$ is $0$ .

Explanation of Solution

Given information:

The given limit is $\underset{y\to -3}{\lim }\left(\frac{y^2+4y+3}{y^2-3}\right)$ .

Calculation:

Substitute $-3$ for $y$ in the given limit.

  $\underset{y\to -3}{\lim }\left(\frac{y^2+4y+3}{y^2-3}\right)=\frac{{\left(-3\right)}^2+4\times \left(-3\right)+3}{\begin{array}{l}{\left(-3\right)}^2-3\\ =\frac{9-12+3}{9-3}\\ =0\end{array}}$

Therefore, the value of $\underset{y\to -3}{\lim }\left(\frac{y^2+4y+3}{y^2-3}\right)$ is $0$ .

Check the limit graphically:

Follow the steps using graphing calculator to graph a function $f\left(y\right)=\frac{y^2+4y+3}{y^2-3}$ .

First press “ON” button on graphical calculator. Press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $f\left(y\right)=\frac{y^2+4y+3}{y^2-3}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{+}\boxed{4}\boxed{\text{X}}\boxed{+}\boxed{3}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{-}\boxed{3}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\text{X}^2+4\text{X}+3\right)/\left(\text{X}^2-3\right)$

Press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function. Enter the keystrokes for the value of the function at particular point $\boxed{2\text{nd}}\boxed{\text{TRACE}}\boxed{1}\boxed{(-)}\boxed{3}$ and press the $\boxed{\text{ENTER}}$ key.

The graph plotted in standard window as shown in Figure (1).

  

Figure (1)

It is verified graphically that the value of $\underset{y\to -3}{\lim }\left(\frac{y^2+4y+3}{y^2-3}\right)$ is $0$ .