Certainly! Let's find the limit:
[tex]\[
\lim_{x \rightarrow 4}\left(\frac{4-x}{2-\sqrt{x}}\right)
\][/tex]
### Step-by-Step Solution:
1. Identify the Type of Limit:
As [tex]\( x \)[/tex] approaches 4, let's examine the expression [tex]\( \frac{4-x}{2-\sqrt{x}} \)[/tex]:
[tex]\[
\frac{4-x}{2-\sqrt{x}}
\][/tex]
Substitute [tex]\( x = 4 \)[/tex] into the expression:
[tex]\[
\text{Numerator: } 4 - 4 = 0
\][/tex]
[tex]\[
\text{Denominator: } 2 - \sqrt{4} = 2 - 2 = 0
\][/tex]
So, we get a [tex]\(\frac{0}{0}\)[/tex] indeterminate form.
2. Simplify the Expression:
To eliminate the indeterminate form, we'll rationalize the denominator. Multiply the numerator and the denominator by the conjugate of the denominator [tex]\(2 + \sqrt{x}\)[/tex].
[tex]\[
\frac{4-x}{2-\sqrt{x}} \times \frac{2+\sqrt{x}}{2+\sqrt{x}} = \frac{(4-x)(2+\sqrt{x})}{(2-\sqrt{x})(2+\sqrt{x})}
\][/tex]
3. Simplify the Denominator:
The denominator simplifies because it's a difference of squares:
[tex]\[
(2-\sqrt{x})(2+\sqrt{x}) = 4 - x
\][/tex]
4. Simplify the Fraction:
Now, the expression becomes:
[tex]\[
\frac{(4-x)(2+\sqrt{x})}{4-x}
\][/tex]
Since [tex]\(4 - x \neq 0\)[/tex] when [tex]\(x \neq 4\)[/tex], we can cancel out the [tex]\(4-x\)[/tex] term in the numerator and denominator:
[tex]\[
2 + \sqrt{x}
\][/tex]
5. Evaluate the Limit:
Now that the expression is simplified, substitute [tex]\( x = 4 \)[/tex]:
[tex]\[
2 + \sqrt{4} = 2 + 2 = 4
\][/tex]
Hence, the limit is:
[tex]\[
\lim_{x \rightarrow 4}\left(\frac{4-x}{2-\sqrt{x}}\right) = 4
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{4}
\][/tex]
