Alright, let's evaluate the limit of the function [tex]\(\frac{\sqrt{x+7}-2}{x+3}\)[/tex] as [tex]\(x\)[/tex] approaches [tex]\(-3\)[/tex].
1. Substitute [tex]\(x = -3\)[/tex] into the function:
[tex]\[\frac{\sqrt{-3+7} - 2}{-3 + 3} = \frac{\sqrt{4} - 2}{0}\][/tex]
Simplifying inside the fraction,
[tex]\[\frac{2 - 2}{0} = \frac{0}{0}\][/tex]
This results in an indeterminate form, [tex]\(\frac{0}{0}\)[/tex]. To resolve this, we need to simplify the expression further.
2. Rationalizing the numerator:
To simplify [tex]\(\frac{\sqrt{x+7} - 2}{x+3}\)[/tex], we multiply the numerator and the denominator by the conjugate of the numerator, [tex]\(\sqrt{x+7} + 2\)[/tex]:
[tex]\[
\frac{\sqrt{x+7} - 2}{x+3} \cdot \frac{\sqrt{x+7} + 2}{\sqrt{x+7} + 2}
\][/tex]
3. Simplify the result:
When we multiply the numerators and the denominators, we get:
[tex]\[
\frac{(\sqrt{x+7} - 2)(\sqrt{x+7} + 2)}{(x+3)(\sqrt{x+7} + 2)}
\][/tex]
The numerator becomes a difference of squares:
[tex]\[
(\sqrt{x+7})^2 - 2^2 = (x + 7) - 4 = x + 3
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
\frac{x+3}{(x+3)(\sqrt{x+7} + 2)} = \frac{1}{\sqrt{x+7} + 2}
\][/tex]
4. Take the limit:
Now, we find the limit as [tex]\(x\)[/tex] approaches [tex]\(-3\)[/tex]:
[tex]\[
\lim_{x \to -3} \frac{1}{\sqrt{x+7} + 2}
\][/tex]
Substitute [tex]\(x = -3\)[/tex] into the simplified expression:
[tex]\[
\frac{1}{\sqrt{-3 + 7} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\][/tex]
Therefore, the limit of [tex]\(\frac{\sqrt{x+7}-2}{x+3}\)[/tex] as [tex]\(x\)[/tex] approaches [tex]\(-3\)[/tex] is [tex]\(\boxed{\frac{1}{4}}\)[/tex].
