What is the numerical value of the limit \lim_{x \to 4} \frac{x - 4}{\sqrt{x + 5} - 3}? — JAMB UTME

What is the numerical value of the limit

\lim_{x \to 4} \frac{x - 4}{\sqrt{x + 5} - 3}

The value of the limit is

6

Multiplying both the numerator and the denominator by the conjugate of the denominator

(\sqrt{x + 5} + 3)

allows the factor

(x - 4)

to cancel out, leaving

\sqrt{x + 5} + 3

. Evaluating this expression as

x \to 4

yields

3 + 3 = 6

Check for direct substitution.

Substituting

x = 4

yields

\frac{4 - 4}{\sqrt{4 + 5} - 3} = \frac{0}{3 - 3} = \frac{0}{0}

, which is an indeterminate form.

Direct substitution gives

\frac{0}{0}

, requiring algebraic simplification such as rationalization.

Rationalize the denominator by multiplying the numerator and denominator by the conjugate

(\sqrt{x + 5} + 3)

\lim_{x \to 4} \frac{(x - 4)(\sqrt{x + 5} + 3)}{(\sqrt{x + 5} - 3)(\sqrt{x + 5} + 3)} = \lim_{x \to 4} \frac{(x - 4)(\sqrt{x + 5} + 3)}{(x + 5) - 9}

Using the difference of squares formula

(a - b)(a + b) = a^2 - b^2

eliminates the square root in the denominator.

Simplify the denominator and cancel out the common factor

(x - 4)

\lim_{x \to 4} \frac{(x - 4)(\sqrt{x + 5} + 3)}{x - 4} = \lim_{x \to 4} (\sqrt{x + 5} + 3)

Since

x \neq 4

when evaluating the limit, the indeterminate factor

(x - 4)

cancels out.

Substitute

x = 4

into the simplified expression.

\sqrt{4 + 5} + 3 = \sqrt{9} + 3 = 3 + 3 = 6

Evaluates the limit after removing the zero-denominator condition.

Limits of indeterminate forms

\frac{0}{0}

involving radicals (Rationalization Technique)

Estimated Time:1m 30s

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