Use algebra and the properties of limits as needed to find the given limit. If the limit does not​ exist, say so.
ModifyingBelow lim With x right arrow 7StartFraction x squared plus 3 x minus 70 Over x squared minus 49 EndFraction

To find the limit of the given function as \( x \) approaches 7, we start by simplifying the expression:

\[
\lim_{{x \to 7}} \frac{x^2 + 3x - 70}{x^2 - 49}
\]

First, we factor both the numerator and the denominator.

1. Factor the numerator \( x^2 + 3x - 70 \):

\[
x^2 + 3x - 70 = (x + 10)(x - 7)
\]

2. Factor the denominator \( x^2 - 49 \):

\[
x^2 - 49 = (x + 7)(x - 7)
\]

Now we rewrite the limit using these factorizations:

\[
\lim_{{x \to 7}} \frac{(x + 10)(x - 7)}{(x + 7)(x - 7)}
\]

Notice that \( (x - 7) \) is a common factor in both the numerator and the denominator. As long as \( x \neq 7 \), we can cancel this common factor:

\[
\lim_{{x \to 7}} \frac{x + 10}{x + 7}
\]

Now, we can directly substitute \( x = 7 \) into the simplified expression:

\[
\frac{7 + 10}{7 + 7} = \frac{17}{14}
\]

Therefore, the limit is:

\[
\lim_{{x \to 7}} \frac{x^2 + 3x - 70}{x^2 - 49} = \frac{17}{14}
\]

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask:
1. How do you simplify expressions with common factors?
2. What are the properties of limits that are most useful in solving problems?
3. Can you explain how to factor quadratic expressions?
4. What happens if the limit does not exist?
5. How do you handle limits involving infinity?

**Tip:** Always check for common factors that can be canceled before directly substituting values in limit problems to simplify the computation process.

Find the limit of the rational function (x^2 + 3x - 70) / (x^2 - 49) as x approaches 7. This detailed solution covers factoring expressions and applying limit properties. Ideal for students in Grades 9-12.

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