Chapter 3.1, Problem 3QR

To determine

To calculate: The value of the limit algebraically.

Answer to Problem 3QR

The required value of the limit is $-1$ .

Explanation of Solution

Given information:

The limit $=\underset{y\to 0^{-}}{\lim }\frac{\left|y\right|}{y}$

Calculation:

In the given limit, the symbol of 0 is negative;

So, the value of the limit can be evaluated when $y$ approaches 0 from the negative numbers.

This means we can write the term $\left|y\right|$ as $-y$ .

  $\underset{y\to 0^{-}}{\lim }\frac{\left|y\right|}{y}=\underset{y\to 0^{-}}{\lim }\frac{-y}{y}$

It is seen that the limit becomes undefined at $y=0$ as the denominator becomes 0.

To find the value of the limit, first eliminate the denominator.

Divide the numerator and the denominator by $y$ .

  $\begin{array}{c}\underset{y\to 0^{-}}{\lim }\frac{\frac{-y}{y}}{\frac{y}{y}}=\underset{y\to 0^{-}}{\lim }\frac{-1}{1}\\ =\underset{y\to 0^{-}}{\lim }\left(-1\right)\end{array}$

Since, the y = 0 cannot be substituted in the above expression;

So, the value of the limit is $-1$ .

Hence, the required value of the limit is $-1$ .