Questions: Find the following limit. Answer as a fraction or integer (no decimals.) lim as x approaches 2 of (x^2+9x-22)/(x^2-4) = □

$Find the following limit. Answer as a fraction or integer (no decimals.) lim as x approaches 2 of (x^2+9x-22)/(x^2-4) = □$

Transcript text: Find the following limit. Answer as a fraction or integer (no decimals.) \[ \lim _{x \rightarrow 2} \frac{x^{2}+9 x-22}{x^{2}-4}=\square \]

Solution

Solution Steps

Step 1: Substitute the limit value into the function

Substitute $ x = 2 $ into the numerator and denominator: \[ \text{Numerator: } 2^{2} + 9(2) - 22 = 4 + 18 - 22 = 0 \] \[ \text{Denominator: } 2^{2} - 4 = 4 - 4 = 0 \] Since both the numerator and denominator evaluate to 0, the limit is in an indeterminate form $ \frac{0}{0} $.

Step 2: Factor the numerator and denominator

Factor the numerator and denominator to simplify the expression: \[ \text{Numerator: } x^{2} + 9x - 22 = (x + 11)(x - 2) \] \[ \text{Denominator: } x^{2} - 4 = (x + 2)(x - 2) \]

Step 3: Simplify the expression

Cancel the common factor $ (x - 2) $ from the numerator and denominator: \[ \frac{(x + 11)(x - 2)}{(x + 2)(x - 2)} = \frac{x + 11}{x + 2} \]

Step 4: Re-evaluate the limit

Substitute $ x = 2 $ into the simplified expression: \[ \frac{2 + 11}{2 + 2} = \frac{13}{4} \]