Math / Calculus

QuestionWho did it? $\lim _{x \rightarrow 2} \frac{x^{2}+2 x-8}{x-2}=$
0 = Selena Gomez 2 = Lebron James 6 = Taylor Swift DNE = Jay Z

Studdy Solution

STEP 1

1. The expression is a limit problem involving a rational function.
2. Direct substitution initially results in an indeterminate form.
3. Simplification or algebraic manipulation is necessary to evaluate the limit.

STEP 2

1. Identify the indeterminate form.
2. Simplify the expression.
3. Evaluate the limit after simplification.

STEP 3

First, substitute $x = 2$ directly into the expression to check for an indeterminate form:
$\frac{2^{2} + 2 \cdot 2 - 8}{2 - 2} = \frac{4 + 4 - 8}{0} = \frac{0}{0}$
Since this results in the indeterminate form $\frac{0}{0}$ , we need to simplify the expression.

STEP 4

Factor the numerator $x^2 + 2x - 8$ to simplify the expression:
$x^2 + 2x - 8 = (x - 2)(x + 4)$
Rewrite the original expression using this factorization:
$\frac{x^2 + 2x - 8}{x - 2} = \frac{(x - 2)(x + 4)}{x - 2}$
Cancel the common factor $x - 2$ from the numerator and the denominator:
$\frac{(x - 2)(x + 4)}{x - 2} = x + 4 \quad \text{for} \quad x \neq 2$

STEP 5

Now evaluate the limit of the simplified expression as $x$ approaches 2:
$\lim_{x \to 2} (x + 4) = 2 + 4 = 6$
The limit evaluates to 6, which corresponds to Taylor Swift.
$\boxed{6}$

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