Evaluate the following limit.
 
Enter the exact answer. If the limit does not exist, enter DNE.
 
limx→4x+5−−−−√−3x2−x−12=limx→4x+5−3x2-x−12=


To evaluate the limit

\[
\lim_{x \to 4} \frac{\sqrt{x+5} - 3}{x^2 - x - 12}
\]

we first try direct substitution of \(x = 4\):

\[
\text{Numerator: } \sqrt{4+5} - 3 = \sqrt{9} - 3 = 3 - 3 = 0
\]

\[
\text{Denominator: } 4^2 - 4 - 12 = 16 - 4 - 12 = 0
\]

Since both the numerator and denominator result in 0, we have an indeterminate form \( \frac{0}{0} \). To resolve this, we can apply L'Hôpital's Rule, but before doing so, let's try factoring or simplifying.

### Step 1: Simplifying the Denominator
The denominator can be factored as follows:

\[
x^2 - x - 12 = (x - 4)(x + 3)
\]

So the limit becomes:

\[
\lim_{x \to 4} \frac{\sqrt{x+5} - 3}{(x - 4)(x + 3)}
\]

### Step 2: Rationalizing the Numerator
We can try rationalizing the numerator by multiplying the numerator and denominator by the conjugate of the numerator:

\[
\frac{\sqrt{x+5} - 3}{(x - 4)(x + 3)} \cdot \frac{\sqrt{x+5} + 3}{\sqrt{x+5} + 3} = \frac{(\sqrt{x+5})^2 - 3^2}{(x - 4)(x + 3)(\sqrt{x+5} + 3)}
\]

Simplifying the numerator:

\[
(\sqrt{x+5})^2 - 3^2 = (x+5) - 9 = x - 4
\]

So the limit simplifies to:

\[
\lim_{x \to 4} \frac{x - 4}{(x - 4)(x + 3)(\sqrt{x+5} + 3)} = \lim_{x \to 4} \frac{1}{(x + 3)(\sqrt{x+5} + 3)}
\]

### Step 3: Direct Substitution
Now, we can substitute \(x = 4\) into the simplified expression:

\[
\frac{1}{(4 + 3)(\sqrt{4 + 5} + 3)} = \frac{1}{7 \times (3 + 3)} = \frac{1}{7 \times 6} = \frac{1}{42}
\]

Thus, the limit is:

\[
\boxed{\frac{1}{42}}
\]

Would you like any further details or explanations? Here are some related questions:

1. How do you apply L'Hôpital's Rule to indeterminate forms?
2. What is the conjugate of a binomial, and how is it used in limits?
3. How do you factor quadratic expressions like \(x^2 - x - 12\)?
4. When is it appropriate to rationalize the numerator in a limit problem?
5. What are other techniques to solve indeterminate forms in limits?

**Tip:** Rationalizing the numerator is a useful strategy when dealing with square roots in limits.

Learn how to evaluate the limit lim(x->4) (sqrt(x+5) - 3) / (x^2 - x - 12) step-by-step with detailed explanations. This problem involves limits, square roots, and factoring quadratics. Suitable for advanced high school students studying calculus.

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