Question

Evaluate the limit: $$ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{x^2 - 3x - 4} \right) $$ is equal to 

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{2} \)
• E. \( 1 \)

Hint:

When evaluating limits involving rational expressions, always try factoring the denominator first to check for simplifications.

Answer 1

The Correct Option is B

Given: \[ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{x^2 - 3x - 4} \right) \] First, factorize the denominator in the second fraction: \[ x^2 - 3x - 4 = (x - 4)(x + 1) \] Thus, rewriting the expression: \[ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{(x - 4)(x + 1)} \right) \] Taking the common denominator: \[ \frac{(x+1) - 5}{(x-4)(x+1)} \] \[ \frac{x + 1 - 5}{(x - 4)(x + 1)} \] \[ \frac{x - 4}{(x - 4)(x + 1)} \] Cancel \( (x-4) \) from numerator and denominator: \[ \lim_{x \to 4} \frac{1}{x + 1} \] Substituting \( x = 4 \): \[ \frac{1}{4 + 1} = \frac{1}{5} \] Thus, the correct answer is (B) \( \frac{1}{5} \).