QuestionAugust 25, 2025

# underline ( )lim _(xarrow 4)((sqrt (x)-2)/(x-4))

# underline ( )lim _(xarrow 4)((sqrt (x)-2)/(x-4))
# underline ( )lim _(xarrow 4)((sqrt (x)-2)/(x-4))

Solution
4.6(220 votes)

Answer

\(\frac{1}{4}\) Explanation 1. Identify Indeterminate Form Substitute x = 4 into \frac{\sqrt{x} - 2}{x - 4} to check for indeterminacy. It results in \frac{0}{0}, indicating an indeterminate form. 2. Rationalize the Numerator Multiply numerator and denominator by the conjugate: \frac{\sqrt{x} - 2}{x - 4} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}. 3. Simplify Expression The numerator becomes (\sqrt{x})^2 - 2^2 = x - 4. The expression simplifies to \frac{x - 4}{(x - 4)(\sqrt{x} + 2)} = \frac{1}{\sqrt{x} + 2}. 4. Evaluate the Limit Substitute x = 4 into \frac{1}{\sqrt{x} + 2}, resulting in \frac{1}{4}.

Explanation

1. Identify Indeterminate Form<br /> Substitute $x = 4$ into $\frac{\sqrt{x} - 2}{x - 4}$ to check for indeterminacy. It results in $\frac{0}{0}$, indicating an indeterminate form.<br /><br />2. Rationalize the Numerator<br /> Multiply numerator and denominator by the conjugate: $\frac{\sqrt{x} - 2}{x - 4} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}$.<br /><br />3. Simplify Expression<br /> The numerator becomes $(\sqrt{x})^2 - 2^2 = x - 4$. The expression simplifies to $\frac{x - 4}{(x - 4)(\sqrt{x} + 2)} = \frac{1}{\sqrt{x} + 2}$.<br /><br />4. Evaluate the Limit<br /> Substitute $x = 4$ into $\frac{1}{\sqrt{x} + 2}$, resulting in $\frac{1}{4}$.
Click to rate:

Similar Questions