Question

How do you find a one sided limit for an absolute value function?

Answer 1

Hint : A function is expressed in terms of a variable quantity represented by an alphabet, as the value of the variable changes; the value of the function also changes. The values of the variable are called the input or index values. Thus the limit of a function is defined as the value that the function approaches when the input or index value approaches some value. As the input value approaches some specific value from the left or the right side, both these limits are called one-sided limits of the function. Thus, we have to find the left-hand limit and right-hand limit of the absolute value function.

Complete step-by-step answer :
Let an absolute value function be $\left| {x + 1} \right|$ , the function is changes sign from negative to positive at x=-1, so for finding one-sided limit, its limit is expressed as –
$\mathop {\lim }\limits_{x \to - 1} \left| {x + 1} \right|$
Left-hand limit –
$\mathop {\lim }\limits_{x \to - {1^ - }} = - (x + 1) \\\ \Rightarrow \mathop {\lim }\limits_{x \to - 1 - h} - x - 1,\,where\,h \to 0 \\\ \Rightarrow \mathop {\lim }\limits_{h \to 0} - ( - 1 - h) - 1 \\\ \Rightarrow \mathop {\lim }\limits_{x \to - {1^ - }} - x = \mathop {\lim }\limits_{h \to 0} 1 + h - 1 = 0 \\\$
Right-hand limit –
$\mathop {\lim }\limits_{x \to - {1^ + }} (x + 1) \\\ \Rightarrow \mathop {\lim }\limits_{x \to - 1 + h} x + 1,\,where\,h \to 0 \\\ \Rightarrow \mathop {\lim }\limits_{h \to 0} - 1 + h + 1 \\\ \Rightarrow \mathop {\lim }\limits_{x \to - {1^ + }} = \mathop {\lim }\limits_{h \to 0} h = 0 \\\$
Now, Left-hand limit is equal to the right-hand limit.
This way we find one side limit for an absolute value function.

Note : An absolute value function always gives a non-negative number as the answer, that is, if the input value is a negative number, the absolute value function (also called modulus function) converts it into a positive function while a positive input value remains a positive value. $\left| x \right| = x$ when x is positive and $\left| x \right| = - x$ when x is negative; thus an absolute is a piecewise function that is it can be written as two subfunctions, that’s why we can find its one-sided value.