Questions: Find the limit. (If an answer does not exist, enter DNE.) lim as x approaches 2 of (x^2+x-6)/(x^2-4)

Find the limit. (If an answer does not exist, enter DNE.) lim as x approaches 2 of (x^2+x-6)/(x^2-4)
Transcript text: Find the limit. (If an answer does not exist, enter DNE.) \[ \lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x^{2}-4} \] Need Help? Read It
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Solution

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Solution Steps

To find the limit of the given function as \( x \) approaches 2, we can start by simplifying the expression. We will factor both the numerator and the denominator and then cancel out any common factors. After simplification, we can directly substitute \( x = 2 \) to find the limit.

Step 1: Factor the Numerator and Denominator

We start by factoring the numerator and the denominator of the given function: \[ \frac{x^2 + x - 6}{x^2 - 4} \]

The numerator \(x^2 + x - 6\) factors to: \[ (x - 2)(x + 3) \]

The denominator \(x^2 - 4\) factors to: \[ (x - 2)(x + 2) \]

Step 2: Simplify the Expression

Next, we simplify the expression by canceling out the common factor \((x - 2)\): \[ \frac{(x - 2)(x + 3)}{(x - 2)(x + 2)} = \frac{x + 3}{x + 2} \]

Step 3: Substitute \(x = 2\) into the Simplified Expression

We then substitute \(x = 2\) into the simplified expression: \[ \frac{x + 3}{x + 2} \bigg|_{x = 2} = \frac{2 + 3}{2 + 2} = \frac{5}{4} \]

Final Answer

\(\boxed{\frac{5}{4}}\)

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