Question

In Exercises 29 and 30 , use a graph to show that the limit does not exist. $$\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 4 } { x - 1 }$$

Short answer

The limit of the given function as \(x\) approaches 1 does not exist because the right-hand limit and the left-hand limit are not equal.

Step-by-step solution

Simplify the expression

Before evaluating the limit, simplify the expression. The function \(\frac { x ^ { 2 } - 4 } { x - 1 }\) can be written as \(\frac { (x + 2)(x - 2) } { x - 1 }\), where this simplification is done by using the difference of squares formula in the numerator.

Calculate the right-hand limit

Evaluate the limit of the function as \(x\) approaches 1 from the positive (right) side. This is given by \(\lim_ { x \rightarrow 1^+ } \frac { (x + 2)(x - 2) } { x - 1 }\). As \(x\) approaches 1 from the right side, the limit is \(3(\$x\rightarrow 1^{+}\$) - 1 = 2.

Calculate the left-hand limit

Evaluate the limit of the function as \(x\) approaches 1 from the negative (left) side. This is given by \(\lim_{x\rightarrow 1^-} \frac { (x + 2)(x - 2) } { x - 1 }\). As \(x\) approaches 1 from the left side, the limit is \(-1(\$x\rightarrow 1^{-}\$) - 1 = -2.

Compare the right-hand and left-hand limits

The right-hand limit (2) and the left-hand limit (-2) are not equal. When the two one-sided limits are not the same, the general limit does not exist.

Represent it graphically

Plotting the function \(\frac { (x + 2)(x - 2) } { x - 1 }\) on a graph will show a gap or discontinuity at \(x = 1\). The value from the right side will be different from the value on the left, confirming visually that the limit does not exist at \(x = 1\).