SOLUTION: Limit Theorem Worksheet with Answers - Studypool - Free Printable

You've uploaded an image containing three limit problems, each solved using Limit Theorem 5, which states:

> Limit of the Sum and Difference of n Functions:
> If $ \lim_{x \to a} f(x) $ and $ \lim_{x \to a} g(x) $ exist, then:
> $$
> \lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)
> $$

This theorem allows us to evaluate limits of polynomial functions by direct substitution, since polynomials are continuous everywhere.

Let’s go through each problem step-by-step and verify the solutions.

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Problem 3:

$$
\lim_{x \to -2} (x^2 + 3x - 7)
$$

Step-by-step solution:

We substitute $ x = -2 $ into the expression:
$$
(-2)^2 + 3(-2) - 7 = 4 - 6 - 7 = -9
$$

✔ Answer: $-9$ — Correct.

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Problem 4:

$$
\lim_{x \to 0} (5x^2 - x + 10)
$$

Substitute $ x = 0 $:
$$
5(0)^2 - (0) + 10 = 0 - 0 + 10 = 10
$$

✔ Answer: $10$ — Correct.

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Problem 5:

$$
\lim_{r \to -1} (2r^3 - r^2 - 6r + 9)
$$

Substitute $ r = -1 $:
$$
2(-1)^3 - (-1)^2 - 6(-1) + 9 = 2(-1) - (1) + 6 + 9 = -2 - 1 + 6 + 9
$$
$$
= (-3) + 6 + 9 = 3 + 9 = 12
$$

✔ Answer: $12$ — Correct.

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✔ Final Verification:

All three limits were evaluated correctly using direct substitution, which is valid because:
- All functions are polynomials, and
- Polynomials are continuous everywhere, so $ \lim_{x \to a} f(x) = f(a) $

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Summary of Answers:

1. $ \lim_{x \to -2} (x^2 + 3x - 7) = \boxed{-9} $
2. $ \lim_{x \to 0} (5x^2 - x + 10) = \boxed{10} $
3. $ \lim_{r \to -1} (2r^3 - r^2 - 6r + 9) = \boxed{12} $

All answers in your image are correct and properly justified using Limit Theorem 5 (sum/difference rule). Great work!