Questions: Evaluate the limit: lim as x approaches 6 of (x^2 + 2x - 48)/(x - 6)

Transcript text: Evaluate the limit: $\lim _{x \rightarrow 6} \frac{x^{2}+2 x-48}{x-6}$

Solution

Solution Steps

To evaluate the limit, we first check if direct substitution results in an indeterminate form. If it does, we attempt to simplify the expression, often by factoring, to cancel out the problematic term. Once simplified, we can substitute the value to find the limit.

Step 1: Check for Indeterminate Form

To evaluate the limit $\lim _{x \rightarrow 6} \frac{x^{2}+2x-48}{x-6}$, we first substitute $x = 6$ into the expression. This results in the indeterminate form $\frac{0}{0}$, indicating that further simplification is needed.

Step 2: Factor the Numerator

The numerator $x^2 + 2x - 48$ can be factored. Factoring gives us $(x - 6)(x + 8)$.

Step 3: Simplify the Expression

Substitute the factored form back into the original expression: \[ \frac{(x - 6)(x + 8)}{x - 6} \] Cancel the common factor $(x - 6)$ from the numerator and the denominator: \[ x + 8 \]

Step 4: Evaluate the Limit

Now, substitute $x = 6$ into the simplified expression $x + 8$: \[ 6 + 8 = 14 \]