https://r2.erweima.ai/mathbot/image/1725665508012jmcp5e52.jpg

The image contains several calculus exercises involving limits. Let's go through each problem one by one.

### **Exercise 16**
These limits should be solved using L'Hôpital's Rule.

#### a) \( \lim_{x \to 0} \frac{x^3 - \sin x}{x} \)
This limit is indeterminate in the form \( \frac{0}{0} \). We can apply L'Hôpital's Rule, which states that if the limit is indeterminate, we can differentiate the numerator and denominator until the indeterminate form is resolved.

#### b) \( \lim_{x \to 0} \left( \frac{\ln(1 + x)}{x} \right)^{\frac{1}{x}} \)
This requires using logarithmic properties and possibly L'Hôpital's Rule, as the expression involves both logarithms and a limit in indeterminate form.

---

### **Exercise 17**
These are basic limit problems that require direct computation.

#### a) \( \lim_{x \to 0} \frac{\cos x - 1}{x^2} \)
This limit also appears to be indeterminate in the form \( \frac{0}{0} \), so L'Hôpital's Rule might apply. We could also use series expansions for \( \cos x \) around \( x = 0 \).

#### b) \( \lim_{x \to 0} \frac{\cos(2x)}{x^2} \)
Another limit with an indeterminate form. L'Hôpital's Rule or the Taylor series expansion of \( \cos(2x) \) may be useful.

---

### **Exercise 18**
These are also limit problems involving trigonometric functions.

#### a) \( \lim_{x \to 0} \frac{\tan x \cdot \cos x - \sin x}{x^2} \)
This limit likely involves applying both L'Hôpital's Rule and trigonometric identities to resolve.

#### b) \( \lim_{x \to 0} \frac{\ln(1 + x)}{x} \)
This is a standard limit that can be resolved using L'Hôpital's Rule or by recognizing it as a common limit expression.

---

Would you like to go through any specific problem in more detail, or solve each one step by step?

Here are 5 relative questions to expand your understanding:
1. How does L'Hôpital's Rule work when applied to indeterminate forms?
2. What is the Taylor series expansion of sine and cosine functions around \( x = 0 \)?
3. Can logarithmic functions always be handled with L'Hôpital's Rule in limits?
4. How can trigonometric identities simplify complex limit problems?
5. What are common limit forms that frequently appear in calculus?

**Tip**: For limit problems involving trigonometric functions, series expansions can often simplify the problem significantly, especially around \( x = 0 \).

Exercise 16: Calculate the limits using L'Hôpital's Rule. Exercise 17: Find the limits. Exercise 18: Calculate the limits.

This page covers solving limit problems using L'Hôpital's Rule and explores limits involving trigonometric and logarithmic functions. Key concepts include L'Hôpital's Rule, Taylor series expansions, and logarithmic properties. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation