Limits Properties — Question 8

Question 8

Assume the following limits exist: $$\lim_{x \to -1} f(x) = 2, \qquad \lim_{x \to -1} g(x) = 3$$

Evaluate the limit: $$\lim_{x \to -1} \left[ (f(x) - g(x))^2 + 4f(x) \right]$$

Use limit laws to justify each step.

Question 8 - Solution

We are given: $$\lim_{x \to -1} f(x) = 2, \qquad \lim_{x \to -1} g(x) = 3$$

We evaluate: $$\lim_{x \to -1} \left[ (f(x) - g(x))^2 + 4f(x) \right]$$

Step 1: Apply limit laws to each part

First, consider the difference: $$\lim_{x \to -1} (f(x) - g(x)) = \lim_{x \to -1} f(x) - \lim_{x \to -1} g(x) = 2 - 3 = -1$$

Now square the result: $$\lim_{x \to -1} (f(x) - g(x))^2 = (-1)^2 = 1$$

Next, evaluate the linear term: $$\lim_{x \to -1} 4f(x) = 4 \cdot \lim_{x \to -1} f(x) = 4 \cdot 2 = 8$$

Step 2: Combine results

Using the sum rule for limits: $$\lim_{x \to -1} \left[ (f(x) - g(x))^2 + 4f(x) \right] = 1 + 8 = \boxed{9}$$