https://file.aiquickdraw.com/mathbot/file/1725221426706bm3ocvm9.jpg

To solve the limit problem given in the image, we have the following expression:

\[
\lim_{{x \to 7}} \frac{x^2 + 7x}{x^2 - 2x - 63}
\]

### Step 1: Factor the denominator
First, let's factor the denominator \(x^2 - 2x - 63\):

The quadratic equation is in the form \(ax^2 + bx + c\). We need to find two numbers that multiply to \(-63\) (the constant term) and add to \(-2\) (the coefficient of \(x\)).

The numbers \(-9\) and \(7\) satisfy this condition:

\[
x^2 - 2x - 63 = (x - 9)(x + 7)
\]

### Step 2: Simplify the expression
The limit now becomes:

\[
\lim_{{x \to 7}} \frac{x^2 + 7x}{(x - 9)(x + 7)}
\]

The numerator \(x^2 + 7x\) can also be factored:

\[
x^2 + 7x = x(x + 7)
\]

So the limit simplifies to:

\[
\lim_{{x \to 7}} \frac{x(x + 7)}{(x - 9)(x + 7)}
\]

Now, we can cancel out the common factor \((x + 7)\) in the numerator and denominator:

\[
\lim_{{x \to 7}} \frac{x}{x - 9}
\]

### Step 3: Substitute \(x = 7\)
Now, substitute \(x = 7\) into the simplified expression:

\[
\frac{7}{7 - 9} = \frac{7}{-2} = -\frac{7}{2}
\]

### Final Answer
The limit is:

\[
\lim_{{x \to 7}} \frac{x^2 + 7x}{x^2 - 2x - 63} = -\frac{7}{2}
\]

Would you like more details or have any questions?

Here are some related questions you might find helpful:
1. How do you factor quadratic expressions?
2. What are common limit laws and how do they apply?
3. What is the significance of canceling out terms in a limit problem?
4. How do you handle indeterminate forms in limits?
5. What is L'Hôpital's rule and when can it be applied?

**Tip:** When solving limits, always check for common factors that can simplify the expression, especially when the direct substitution results in an indeterminate form like \(\frac{0}{0}\).

Learn how to solve the limit problem lim(x->7) (x^2 + 7x) / (x^2 - 2x - 63) with a step-by-step solution. This problem involves limits, quadratic equations, and factorization. Suitable for advanced high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation