Alright, let's solve the limit [tex]\( \lim _{x \rightarrow 25} \frac{x-25}{\sqrt{x}-5} \)[/tex].
To find this limit, we observe that directly substituting [tex]\( x = 25 \)[/tex] into the function results in a [tex]\(\frac{0}{0}\)[/tex] indeterminate form, so we need to manipulate the expression algebraically to resolve this form.
1. Substitution and Observing the Indeterminate Form:
[tex]\[
\lim _{x \rightarrow 25} \frac{x-25}{\sqrt{x}-5} \quad \text{(substitute \( x = 25 \))}
\][/tex]
[tex]\[
\frac{25-25}{\sqrt{25}-5} = \frac{0}{0} \quad \text{(indeterminate form)}
\][/tex]
2. Algebraic Manipulation (Factoring the numerator):
Notice that [tex]\( x - 25 \)[/tex] can be factored in a way that relates it to [tex]\( \sqrt{x} \)[/tex]. Let's use the identity:
[tex]\[
x - 25 = (\sqrt{x})^2 - 5^2
\][/tex]
We can factor this as:
[tex]\[
x - 25 = (\sqrt{x} - 5)(\sqrt{x} + 5)
\][/tex]
3. Rewriting the original limit:
[tex]\[
\lim _{x \rightarrow 25} \frac{(\sqrt{x} - 5)(\sqrt{x} + 5)}{\sqrt{x} - 5}
\][/tex]
At this point, we can cancel out the common factor [tex]\(\sqrt{x} - 5\)[/tex] from the numerator and the denominator, given that [tex]\( \sqrt{x} \neq 5 \)[/tex] when [tex]\( x \neq 25 \)[/tex]:
[tex]\[
\lim _{x \rightarrow 25} (\sqrt{x} + 5)
\][/tex]
4. Evaluating the simplified limit:
Now, we can directly substitute [tex]\( x = 25 \)[/tex] into the simplified expression:
[tex]\[
\sqrt{25} + 5 = 5 + 5 = 10
\][/tex]
Therefore, the limit is:
[tex]\[
\lim _{x \rightarrow 25} \frac{x-25}{\sqrt{x}-5} = 10
\][/tex]
