Evaluate the following limit: $lim_{x\to2} \frac {x^2 + 4x - 12}{x^2 - 2x}$

1. Analyze the question

We are asked to evaluate the limit of the rational function $\frac{x^2 + 4x - 12}{x^2 - 2x}$ as $x$ approaches $2$.

2. Explain the concept

When evaluating limits, the first step is direct substitution. If substituting $x = 2$ results in a defined value, that is our answer. However, if it results in $\frac{0}{0}$ (an indeterminate form), we must simplify the expression by factoring the numerator and the denominator to cancel out the common term causing the zero.

3. Show the solution

Step 1: Direct Substitution Substitute $x = 2$ into the expression: Numerator: $(2)^2 + 4(2) - 12 = 4 + 8 - 12 = 0$ Denominator: $(2)^2 - 2(2) = 4 - 4 = 0$ Since we have $\frac{0}{0}$, we must factorize.

Step 2: Factorize the Numerator The numerator is $x^2 + 4x - 12$. We look for two numbers that multiply to $-12$ and add to $+4$. These numbers are $+6$ and $-2$. $x^2 + 4x - 12 = (x + 6)(x - 2)$

Step 3: Factorize the Denominator The denominator is $x^2 - 2x$. We can factor out the common $x$: $x^2 - 2x = x(x - 2)$

Step 4: Simplify the expression Now, rewrite the limit: $\lim_{x\to2} \frac{(x + 6)(x - 2)}{x(x - 2)}$

Cancel the common factor $(x - 2)$: $\lim_{x\to2} \frac{x + 6}{x}$

Step 5: Evaluate the limit Now substitute $x = 2$ into the simplified expression: $\frac{2 + 6}{2} = \frac{8}{2} = 4$

4. Identify the answer

The calculated value is 4, which corresponds to option A.

5. Exam Tip

Whenever you see a limit problem in JAMB that results in $\frac{0}{0}$, don't panic! It is a clear signal that there is a "hidden" factor in both the top and bottom that needs to be cancelled out. Always factorize first before re-substituting.