https://file.aiquickdraw.com/mathbot/file/1726878686943e97beagi.jpg

Let's solve each of the limits step by step based on the given functions \( f(x) = x^3 \) and \( g(x) = \frac{\sqrt{x^2 + 5}}{2x^2} \):

### (a) \( \lim_{x \to 2} f(x) \)

We are given \( f(x) = x^3 \). So,

\[
\lim_{x \to 2} f(x) = \lim_{x \to 2} x^3 = 2^3 = 8
\]

### (b) \( \lim_{x \to 2} g(x) \)

We are given \( g(x) = \frac{\sqrt{x^2 + 5}}{2x^2} \). Now, let's compute the limit:

\[
\lim_{x \to 2} g(x) = \lim_{x \to 2} \frac{\sqrt{x^2 + 5}}{2x^2}
\]
Substitute \( x = 2 \):

\[
g(2) = \frac{\sqrt{2^2 + 5}}{2(2^2)} = \frac{\sqrt{4 + 5}}{2(4)} = \frac{\sqrt{9}}{8} = \frac{3}{8}
\]

### (c) \( \lim_{x \to 2} [f(x)g(x)] \)

Using the results from parts (a) and (b), we can find this limit as the product of the two previous limits:

\[
\lim_{x \to 2} [f(x)g(x)] = \lim_{x \to 2} f(x) \cdot \lim_{x \to 2} g(x)
\]

Substitute the values:

\[
8 \cdot \frac{3}{8} = 3
\]

### (d) \( \lim_{x \to 2} [g(x) - f(x)] \)

Now, subtract the limits from parts (a) and (b):

\[
\lim_{x \to 2} [g(x) - f(x)] = \lim_{x \to 2} g(x) - \lim_{x \to 2} f(x)
\]

Substitute the values:

\[
\frac{3}{8} - 8 = \frac{3}{8} - \frac{64}{8} = \frac{-61}{8}
\]

### Final Answers:
(a) \( \lim_{x \to 2} f(x) = 8 \)

(b) \( \lim_{x \to 2} g(x) = \frac{3}{8} \)

(c) \( \lim_{x \to 2} [f(x)g(x)] = 3 \)

(d) \( \lim_{x \to 2} [g(x) - f(x)] = \frac{-61}{8} \)

---

Would you like more details or clarifications on any of these calculations? Here are five related questions you can explore:

1. What happens to the limit of \( f(x) \) as \( x \to \infty \)?
2. How would you compute \( \lim_{x \to 0} g(x) \)?
3. What techniques could be used if the limits involved indeterminate forms?
4. Can you find the derivative of \( g(x) \)?
5. What is the behavior of \( g(x) \) as \( x \to \infty \)?

**Tip**: When working with products of limits, always check if each individual limit exists first!

Find the following limits given the functions: f(x) = x^3, g(x) = (sqrt(x^2 + 5))/(2x^2). Solve for (a) lim f(x) as x approaches 2, (b) lim g(x) as x approaches 2, (c) lim [f(x)g(x)] as x approaches 2, (d) lim [g(x) - f(x)] as x approaches 2.

Solve limit problems involving the functions f(x) = x^3 and g(x) = (sqrt(x^2 + 5))/(2x^2) at x = 2. Learn how to compute individual limits and apply limit laws for products and differences. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation