Question

The limit of the function \( \lim_{x \to 2} \frac{2x^2 + 2x - 12}{x^2 - 4} \) is .............. (rounded off to 1 decimal)

Hint:

When faced with the indeterminate form \( \frac{0}{0} \), factor both the numerator and denominator to simplify the expression before substituting the value of \( x \).

Answer 1

Step 1: Analyzing the function.
We are given the function: \[ f(x) = \frac{2x^2 + 2x - 12}{x^2 - 4} \] First, we check if substituting \( x = 2 \) directly into the function gives a meaningful value:
Substituting \( x = 2 \): \[ \text{Numerator} = 2(2^2) + 2(2) - 12 = 8 + 4 - 12 = 0 \] \[ \text{Denominator} = (2^2) - 4 = 4 - 4 = 0 \] Since both the numerator and denominator are 0, we have an indeterminate form \( \frac{0}{0} \), so we must simplify the expression.
Step 2: Simplifying the expression.
Factor the numerator and denominator: \[ \text{Numerator:} \, 2x^2 + 2x - 12 = 2(x^2 + x - 6) = 2(x - 2)(x + 3) \] \[ \text{Denominator:} \, x^2 - 4 = (x - 2)(x + 2) \] Now the function becomes: \[ f(x) = \frac{2(x - 2)(x + 3)}{(x - 2)(x + 2)} \] We can cancel out the common factor \( (x - 2) \): \[ f(x) = \frac{2(x + 3)}{x + 2} \] Step 3: Substituting \( x = 2 \) again.
Substitute \( x = 2 \) into the simplified expression: \[ f(2) = \frac{2(2 + 3)}{2 + 2} = \frac{2(5)}{4} = \frac{10}{4} = 2.5 \] Final Answer: \[ \boxed{2.5} \]