Certainly! Let’s find the limit [tex]\(\lim _{x \rightarrow 4} \frac{2-\sqrt{x}}{x-4}\)[/tex].
Step-by-Step Solution:
1. Identify Indeterminate Form:
First, we need to check whether directly substituting [tex]\(x = 4\)[/tex] into the function [tex]\(\frac{2 - \sqrt{x}}{x - 4}\)[/tex] yields an indeterminate form:
[tex]\[
\frac{2 - \sqrt{4}}{4 - 4} = \frac{2 - 2}{0} = \frac{0}{0}
\][/tex]
Since we get a [tex]\(\frac{0}{0}\)[/tex] indeterminate form, we need another method to evaluate the limit.
2. Rationalizing the Numerator:
To clear the indeterminate form, we can try rationalizing the numerator. Multiply the numerator and the denominator by the conjugate of the numerator, [tex]\(2 + \sqrt{x}\)[/tex]:
[tex]\[
\frac{2 - \sqrt{x}}{x - 4} \cdot \frac{2 + \sqrt{x}}{2 + \sqrt{x}}
\][/tex]
This step utilizes the difference of squares formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
[tex]\[
= \frac{(2 - \sqrt{x})(2 + \sqrt{x})}{(x - 4)(2 + \sqrt{x})}
\][/tex]
Simplify the numerator using the difference of squares:
[tex]\[
= \frac{4 - x}{(x - 4)(2 + \sqrt{x})}
\][/tex]
3. Simplify the Expression:
Notice that [tex]\(4 - x = -(x - 4)\)[/tex], so we can write:
[tex]\[
= \frac{-(x - 4)}{(x - 4)(2 + \sqrt{x})}
\][/tex]
Cancel out the [tex]\((x - 4)\)[/tex] terms in the numerator and the denominator:
[tex]\[
= \frac{-1}{2 + \sqrt{x}}
\][/tex]
4. Substitute the Limit Value:
Now, we substitute [tex]\(x = 4\)[/tex] into the simplified expression:
[tex]\[
= \frac{-1}{2 + \sqrt{4}} = \frac{-1}{2 + 2} = \frac{-1}{4}
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{-\frac{1}{4}}
\][/tex]
