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The limit expression given is:

\[
\lim_{x \to a} \frac{(x + a)^2 - 17x^2 + 13ax}{x - a}
\]

To solve this limit, we can apply L'Hôpital's rule if the limit results in an indeterminate form (like \(\frac{0}{0}\)). However, before that, let's try simplifying the expression first.

### Step 1: Expand and Simplify the Numerator

First, expand \((x + a)^2\):

\[
(x + a)^2 = x^2 + 2ax + a^2
\]

Now, substitute this into the numerator:

\[
\text{Numerator} = (x^2 + 2ax + a^2) - 17x^2 + 13ax
\]

Simplify by combining like terms:

\[
x^2 + 2ax + a^2 - 17x^2 + 13ax = (1x^2 - 17x^2) + (2ax + 13ax) + a^2
\]

\[
= -16x^2 + 15ax + a^2
\]

So the expression becomes:

\[
\lim_{x \to a} \frac{-16x^2 + 15ax + a^2}{x - a}
\]

### Step 2: Factorize the Numerator (if possible)

Next, try to factorize the numerator to see if we can simplify the fraction:

Since the expression in the numerator does not factorize easily, let's proceed with L'Hôpital's rule.

### Step 3: Apply L'Hôpital's Rule

L'Hôpital's rule states that if \(\lim_{x \to a} \frac{f(x)}{g(x)}\) is of the indeterminate form \(\frac{0}{0}\), then:

\[
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}
\]

First, differentiate the numerator and denominator:

- The derivative of the numerator \( -16x^2 + 15ax + a^2 \) with respect to \( x \) is:

\[
f'(x) = -32x + 15a
\]

- The derivative of the denominator \( x - a \) with respect to \( x \) is:

\[
g'(x) = 1
\]

So, applying L'Hôpital's rule:

\[
\lim_{x \to a} \frac{-32x + 15a}{1}
\]

### Step 4: Evaluate the Limit

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

\[
= -32a + 15a = -17a
\]

### Final Answer:

The limit is:

\[
\lim_{x \to a} \frac{(x + a)^2 - 17x^2 + 13ax}{x - a} = -17a
\]

Would you like more details on any of these steps or do you have any other questions?

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**Relative Questions:**
1. What is L'Hôpital's rule and when can it be applied?
2. How can you determine if a limit results in an indeterminate form?
3. What are the steps to factorize quadratic expressions?
4. How do you compute the derivative of polynomial functions?
5. What are other methods to evaluate limits if L'Hôpital's rule is not applicable?

**Tip:** When simplifying expressions, always look for opportunities to factor or cancel terms before applying advanced techniques like L'Hôpital's rule. This can often simplify the problem significantly.

Learn how to solve the limit \(\lim_{x \to a} \frac{(x + a)^2 - 17x^2 + 13ax}{x - a}\) using L'Hôpital's Rule with a step-by-step solution. This problem covers limits, polynomial functions, and the application of L'Hôpital's Rule in calculus. Suitable for advanced high school students and college-level learners.

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