https://r2.erweima.ai/mathbot/image/1724091057013uhrj16l3.jpg

The expression in the image is a limit problem. The limit is given by:

\[
\lim_{{x \to 0}} \frac{x}{{4 - \sqrt{x + 16}}}
\]

To solve this limit, let's first simplify the expression. One common approach is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(4 - \sqrt{x + 16}\) is \(4 + \sqrt{x + 16}\).

So, let's rewrite the limit:

\[
\lim_{{x \to 0}} \frac{x}{{4 - \sqrt{x + 16}}} \times \frac{{4 + \sqrt{x + 16}}}{{4 + \sqrt{x + 16}}}
\]

This simplifies to:

\[
\lim_{{x \to 0}} \frac{x \cdot (4 + \sqrt{x + 16})}{{(4 - \sqrt{x + 16})(4 + \sqrt{x + 16})}}
\]

Notice that the denominator now becomes a difference of squares:

\[
(4 - \sqrt{x + 16})(4 + \sqrt{x + 16}) = 16 - (x + 16) = 16 - x - 16 = -x
\]

So the limit becomes:

\[
\lim_{{x \to 0}} \frac{x \cdot (4 + \sqrt{x + 16})}{-x}
\]

We can cancel out the \(x\) from the numerator and denominator (for \(x \neq 0\)):

\[
\lim_{{x \to 0}} \frac{4 + \sqrt{x + 16}}{-1}
\]

This simplifies to:

\[
\lim_{{x \to 0}} -(4 + \sqrt{x + 16})
\]

Now, substitute \(x = 0\) into the expression:

\[
-(4 + \sqrt{0 + 16}) = -(4 + 4) = -8
\]

So, the solution to the limit is:

\[
\boxed{-8}
\]

Would you like any further details or explanations?

Here are some related questions:
1. What is the significance of using the conjugate to simplify limits?
2. How do we handle indeterminate forms in limit problems?
3. What are other techniques to solve limit problems besides using the conjugate?
4. How can you apply L'Hôpital's rule to a limit problem?
5. What are the conditions under which limits do not exist?

**Tip:** When simplifying limit expressions, multiplying by the conjugate can help eliminate square roots and simplify the expression.

Explore how to solve a limit problem using the conjugate method step-by-step. This example focuses on simplifying expressions involving square roots and applying limit rules effectively.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation