Evaluate the limit \lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}. What is the numerical value … — JAMB UTME

Evaluate the limit $\lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}$ . What is the numerical value of this limit?

Cevap: 0.25

The numerical value of the limit is 0.25 (or

\frac{1}{4}

When direct substitution into

\frac{\sqrt{x + 4} - 2}{x}

yields the indeterminate form

\frac{0}{0}

, rationalizing the numerator by multiplying by its conjugate

\sqrt{x + 4} + 2

simplifies the expression to

\frac{1}{\sqrt{x + 4} + 2}

. Taking the limit as

x \to 0

gives

\frac{1}{4} = 0.25

Check direct substitution

Indeterminate form

\frac{0}{0}

Directly evaluating at

x = 0

yields zero in both numerator and denominator.

Multiply by the conjugate of the numerator

\frac{(\sqrt{x + 4} - 2)(\sqrt{x + 4} + 2)}{x(\sqrt{x + 4} + 2)} = \frac{x}{x(\sqrt{x + 4} + 2)}

Rationalizing the radical in the numerator allows cancellation of the term causing the zero denominator.

Cancel common factors and evaluate limit

\frac{1}{\sqrt{0 + 4} + 2} = 0.25

Canceling

x

removes the zero factor, permitting direct evaluation.

Limits of indeterminate algebraic expressions using surd rationalization

Soru