https://file.aiquickdraw.com/mathbot/file/1726629924913g0wyqidv.jpg

The image you provided contains the following limits to evaluate:

Given:
\[
\lim_{x \to +\infty} f(x) = -5, \quad \lim_{x \to +\infty} g(x) = -1, \quad \lim_{x \to +\infty} h(x) = -5,
\]
we need to find the limits for the following expressions:

(a) \(\lim_{x \to +\infty} \left[ f(x) + 8g(x) \right]\)

(b) \(\lim_{x \to +\infty} \left[ f(x)g(x) \right]\)

(c) \(\lim_{x \to +\infty} \frac{7}{g(x)}\)

(d) \(\lim_{x \to +\infty} \frac{6h(x) + 5}{x^2}\)

### Solution:

**(a)** Using the sum rule for limits:
\[
\lim_{x \to +\infty} \left[ f(x) + 8g(x) \right] = \lim_{x \to +\infty} f(x) + 8 \cdot \lim_{x \to +\infty} g(x)
\]
Substitute the given limits:
\[
= -5 + 8(-1) = -5 - 8 = -13
\]

**(b)** Using the product rule for limits:
\[
\lim_{x \to +\infty} \left[ f(x)g(x) \right] = \lim_{x \to +\infty} f(x) \cdot \lim_{x \to +\infty} g(x)
\]
Substitute the given limits:
\[
= (-5) \cdot (-1) = 5
\]

**(c)** Using the limit rule for rational functions:
\[
\lim_{x \to +\infty} \frac{7}{g(x)} = \frac{7}{\lim_{x \to +\infty} g(x)}
\]
Since \(\lim_{x \to +\infty} g(x) = -1\), we get:
\[
\frac{7}{-1} = -7
\]

**(d)** Evaluate the limit:
\[
\lim_{x \to +\infty} \frac{6h(x) + 5}{x^2} = \lim_{x \to +\infty} \frac{6(-5) + 5}{x^2} = \lim_{x \to +\infty} \frac{-30 + 5}{x^2} = \lim_{x \to +\infty} \frac{-25}{x^2}
\]
As \(x^2\) tends to infinity, \(\frac{-25}{x^2} \to 0\). Therefore:
\[
\lim_{x \to +\infty} \frac{-25}{x^2} = 0
\]

### Final Answers:
(a) \(-13\)  
(b) \(5\)  
(c) \(-7\)  
(d) \(0\)

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the sum rule for limits work?
2. Can you apply the product rule for limits to any two functions?
3. Why does the limit of \(\frac{-25}{x^2}\) approach zero as \(x\) approaches infinity?
4. What happens if \(g(x)\) tends to zero in part (c)?
5. What other limit properties can be applied to more complex functions?

**Tip**: Always check for indeterminate forms (like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)) when calculating limits.

Given that lim(x → +∞) f(x) = -5, lim(x → +∞) g(x) = -1, and lim(x → +∞) h(x) = -5, find the following limits: (a) lim(x → +∞) [f(x) + 8g(x)], (b) lim(x → +∞) [f(x)g(x)], (c) lim(x → +∞) 7 / g(x), (d) lim(x → +∞) (6h(x) + 5) / x^2.

Find the limits of expressions involving f(x), g(x), and h(x) as x approaches infinity. Using sum and product rules for limits, evaluate these functions step-by-step. This problem covers advanced algebraic concepts and is suitable for students in Grades 11-12.

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