Evaluate the limit \lim_{x \to 2} \frac{x^2 - 4}{x - 2}. What is the numerical value of th… — JAMB UTME

Evaluate the limit $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ . What is the numerical value of this limit?

Cevap: 4

The value of the limit is

4

Direct substitution of

x = 2

produces the indeterminate form

\frac{0}{0}

. Factoring the numerator gives

\frac{(x-2)(x+2)}{x-2}

. Canceling the non-zero factor

(x-2)

simplifies the expression to

x+2

. Evaluating the limit as

x

approaches

2

yields

2 + 2 = 4

Check the form by direct substitution of

x = 2

Obtained the indeterminate form

\frac{0}{0}

Direct substitution results in division by zero, requiring algebraic simplification.

Factor the polynomial in the numerator

x^2 - 4 = (x - 2)(x + 2)

Difference of two squares factorization allows canceling common terms.

Cancel the common factor

(x - 2)

\frac{(x - 2)(x + 2)}{x - 2} = x + 2

For

x \neq 2

, division by

(x - 2)

is valid.

Evaluate the simplified limit as

x \to 2

2 + 2 = 4

Substitute

x = 2

directly into the continuous polynomial

x + 2

Evaluating indeterminate limits of the form

\frac{0}{0}

via factorization

Soru