To find the limit [tex]\(\lim _{x \rightarrow 3}\left(\frac{x^2-9}{x-3} + \sqrt{x^2+7}\right)\)[/tex], let's break it down step by step.
First, we look at the expression inside the limit:
[tex]\[
\frac{x^2 - 9}{x - 3} + \sqrt{x^2 + 7}
\][/tex]
### Step 1: Simplify the Rational Function [tex]\(\frac{x^2 - 9}{x - 3}\)[/tex]
The term [tex]\(\frac{x^2 - 9}{x - 3}\)[/tex] can be simplified. Notice that [tex]\(x^2 - 9\)[/tex] can be factored as a difference of squares:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
Thus,
[tex]\[
\frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3}
\][/tex]
For [tex]\(x \neq 3\)[/tex], the term [tex]\((x - 3)\)[/tex] in the numerator and denominator cancels out:
[tex]\[
\frac{(x - 3)(x + 3)}{x - 3} = x + 3
\][/tex]
### Step 2: Substitute the Simplified Expression Back
Now the expression becomes:
[tex]\[
x + 3 + \sqrt{x^2 + 7}
\][/tex]
### Step 3: Evaluate the Limit
Now, we need to evaluate the limit of this simplified expression as [tex]\(x\)[/tex] approaches 3:
[tex]\[
\lim_{x \to 3} \left(x + 3 + \sqrt{x^2 + 7}\right)
\][/tex]
Substitute [tex]\(x = 3\)[/tex]:
[tex]\[
3 + 3 + \sqrt{3^2 + 7}
\][/tex]
Simplify inside the square root:
[tex]\[
3 + 3 + \sqrt{9 + 7} = 3 + 3 + \sqrt{16} = 3 + 3 + 4 = 10
\][/tex]
### Conclusion
Therefore,
[tex]\[
\lim _{x \rightarrow 3}\left(\frac{x^2-9}{x-3} + \sqrt{x^2+7}\right) = 10
\][/tex]
