Evaluate the limit \lim_{x \to 3} \frac{x^2 - 5x + 6}{x - 3}. What is the numerical value … — JAMB UTME

Evaluate the limit $\lim_{x \to 3} \frac{x^2 - 5x + 6}{x - 3}$ . What is the numerical value of this limit?

Answer: 1

Answer

The numerical value of the limit is 1.

Substituting x = 3 into the original expression produces the indeterminate form 0/0. Factoring the numerator gives (x - 3)(x - 2). Canceling the common factor (x - 3) yields x - 2. Substituting x = 3 into x - 2 gives 3 - 2 = 1.

Step-by-Step Solution

Check the expression using direct substitution at x = 3.

The substitution yields \frac{3^2 - 5(3) + 6}{3 - 3} = \frac{0}{0}, an indeterminate form.

Direct substitution results in 0/0, indicating that algebraic factorization is needed.

Factorize the quadratic polynomial in the numerator.

x^2 - 5x + 6 = (x - 3)(x - 2)

Finding factors of +6 that sum to -5 allows cancellation of the denominator term.

Cancel the factor (x - 3) and evaluate the remaining linear expression at x = 3.

\lim_{x \to 3} (x - 2) = 3 - 2 = 1

The factor causing zero in the denominator is removed, leaving a continuous function.

Key Concept

Evaluating indeterminate limits (0/0) by algebraic factorization

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic