Question

In Exercises \(31 - 36 ,\) determine the limit. $$\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { | x | }$$

Short answer

The limit of \( \frac { x } { | x | } \) as \( x \) approaches 0 from the left is -1.

Step-by-step solution

Understand the absolute value function

First, understand that for \( x < 0 \) , \( | x | = - x \). This is because the absolute value of a number is its distance from 0 on the number line irrespective of direction. When \( x \) is less than 0, the absolute value is the positive counterpart of \( x \). So, we can rewrite the function as \(\frac { x } { -x } \) for \( x < 0 \).

Simplify the function

Next, simplify the function \(\frac { x } { -x } \). Both numerator and denominator will cancel out leaving -1.

Find the limit of the function

Now we can find the limit of the function as \( x \) approaches 0 from the left side. The function is constant and equal to -1 for \( x < 0 \). Hence, it does not matter what value \( x \) is approaching from the left side, the function will always yield -1. Thus, the limit of the function as \( x \) approaches 0 from the left side is -1.