Limit Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Find limโกxโ†’2x2โˆ’4xโˆ’2\lim_{x \to 2} \frac{x^2 - 4}{x - 2}

Solution

  1. 1
    Direct substitution gives 22โˆ’42โˆ’2=00\frac{2^2 - 4}{2 - 2} = \frac{0}{0}, which is indeterminate.
  2. 2
    Factor the numerator: x2โˆ’4=(xโˆ’2)(x+2)x^2 - 4 = (x - 2)(x + 2).
  3. 3
    Cancel the common factor: (xโˆ’2)(x+2)xโˆ’2=x+2\frac{(x-2)(x+2)}{x-2} = x + 2 for xโ‰ 2x \neq 2.
  4. 4
    Now evaluate: limโกxโ†’2(x+2)=4\lim_{x \to 2} (x + 2) = 4.

Answer

limโกxโ†’2x2โˆ’4xโˆ’2=4\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4
When direct substitution yields 00\frac{0}{0}, try factoring to cancel the common factor. The limit examines behavior near the point, not at the point itself, so cancellation is valid.

About Limit

The value a function gets closer and closer to as the input approaches a specific target value, without necessarily reaching it.

Learn more about Limit โ†’

More Limit Examples

Example 1 easy

Find [formula]

Example 3 hard

Find [formula]

Example 4 easy

Find [formula]

Example 5 medium

Find [formula]

โ† All Limit Examples Limit Concept