1.7 Infinite Limits And Limits At Infinity Homework

1.7 Infinite Limits and Limits at Infinity: A Comprehensive Guide to Homework Problems

Calculus, particularly the study of limits, can often feel like navigating a dense forest. Understanding infinite limits and limits at infinity is crucial for mastering this fundamental concept. This comprehensive guide will illuminate these concepts, providing you with the tools and strategies needed to tackle your homework problems with confidence. We'll delve into the definitions, explore various techniques for evaluating these limits, and work through examples to solidify your understanding.

Understanding the Concepts: Infinite Limits and Limits at Infinity

Before we dive into problem-solving, let's clearly define the terms:

Infinite Limits

An infinite limit describes the behavior of a function as its input approaches a particular value, where the function's output grows without bound (towards positive or negative infinity). We write:

• lim_x→a f(x) = ∞: This means that as x approaches a, the function f(x) becomes arbitrarily large (positive).
• lim_x→a f(x) = -∞: This means that as x approaches a, the function f(x) becomes arbitrarily large (negative).

It's important to note that the limit doesn't actually exist in these cases; the function is diverging. However, the notation indicates the manner of divergence.

Limits at Infinity

A limit at infinity describes the behavior of a function as its input grows without bound (towards positive or negative infinity). We write:

• lim_x→∞ f(x) = L: This means that as x becomes arbitrarily large (positive), the function f(x) approaches the value L.
• lim_x→-∞ f(x) = L: This means that as x becomes arbitrarily large (negative), the function f(x) approaches the value L.

Here, L can be any real number. If the limit exists and equals L, we say the function has a horizontal asymptote at y = L.

Techniques for Evaluating Infinite Limits and Limits at Infinity

Several techniques can help evaluate these limits. Let's examine the most common ones:

1. Direct Substitution

Sometimes, the simplest approach is the most effective. If the function is continuous at the point in question, direct substitution can yield the limit. However, this is rarely the case with infinite limits and limits at infinity.

2. Algebraic Manipulation

Often, algebraic manipulation is required to simplify the expression before evaluating the limit. This might involve factoring, expanding, rationalizing the numerator or denominator, or using other algebraic techniques.

Example: Find lim_x→2 (x² - 4) / (x - 2)

Direct substitution leads to 0/0, an indeterminate form. However, factoring the numerator gives:

lim_x→2 (x - 2)(x + 2) / (x - 2) = lim_x→2 (x + 2) = 4

3. L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. The rule states that if lim_x→a f(x) / g(x) is of the form 0/0 or ∞/∞, then:

lim_x→a f(x) / g(x) = lim_x→a f'(x) / g'(x),

provided the latter limit exists.

Example: Find lim_x→∞ x / e^x

This is of the form ∞/∞. Applying L'Hôpital's Rule repeatedly gives:

lim_x→∞ x / e^x = lim_x→∞ 1 / e^x = 0

4. Analyzing Dominant Terms

For limits at infinity, identifying the dominant terms in the expression is crucial. The dominant terms are the terms that grow the fastest as x approaches infinity. You can often ignore the other terms.

Example: Find lim_x→∞ (3x² + 2x + 1) / (x² - 5)

The dominant terms are 3x² in the numerator and x² in the denominator. Therefore:

lim_x→∞ (3x² + 2x + 1) / (x² - 5) = lim_x→∞ 3x² / x² = 3

5. Squeeze Theorem (or Sandwich Theorem)

The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) for all x near a, and lim_x→a f(x) = lim_x→a h(x) = L, then lim_x→a g(x) = L.

This theorem is particularly useful when dealing with trigonometric functions or expressions involving inequalities.

Working Through Examples: Infinite Limits and Limits at Infinity Homework Problems

Let's tackle some sample problems to solidify your understanding:

Problem 1: Find lim_x→0 1/x²

As x approaches 0, 1/x² becomes arbitrarily large (positive). Therefore:

lim_x→0 1/x² = ∞

Problem 2: Find lim_x→∞ (x³ - 2x + 1) / (2x³ + x²)

The dominant terms are x³ in both the numerator and the denominator. Therefore:

lim_x→∞ (x³ - 2x + 1) / (2x³ + x²) = lim_x→∞ x³ / (2x³) = 1/2

Problem 3: Find lim_x→∞ (√(x² + 1) - x)

This limit is in an indeterminate form (∞ - ∞). To solve it, we rationalize the expression by multiplying by the conjugate:

lim_x→∞ (√(x² + 1) - x) * (√(x² + 1) + x) / (√(x² + 1) + x) = lim_x→∞ (x² + 1 - x²) / (√(x² + 1) + x) = lim_x→∞ 1 / (√(x² + 1) + x) = 0

Problem 4: Find lim_x→0 sin(x) / x

This is an indeterminate form (0/0). Using L'Hôpital's rule:

lim_x→0 sin(x) / x = lim_x→0 cos(x) / 1 = 1

Problem 5: Find lim_x→∞ (x + sin(x)) / x

We can rewrite this as:

lim_x→∞ (1 + sin(x)/x)

Since -1 ≤ sin(x) ≤ 1, -1/x ≤ sin(x)/x ≤ 1/x. As x approaches infinity, both -1/x and 1/x approach 0. By the Squeeze Theorem:

lim_x→∞ sin(x)/x = 0

Therefore, lim_x→∞ (x + sin(x)) / x = 1

Problem 6 (Challenging): Find lim_x→∞ (x^1/x)

This limit requires a clever approach. Let y = x^1/x. Then, ln(y) = (1/x)ln(x). Now we consider the limit of ln(y) as x approaches infinity:

lim_x→∞ (1/x)ln(x) = lim_x→∞ ln(x) / x

This is of the form ∞/∞. Applying L'Hôpital's Rule:

lim_x→∞ ln(x) / x = lim_x→∞ (1/x) / 1 = 0

Since ln(y) → 0 as x → ∞, then y → e⁰ = 1. Therefore:

lim_x→∞ (x^1/x) = 1

Strategies for Success in Your Calculus Homework

• Master the Definitions: Ensure you thoroughly understand the definitions of infinite limits and limits at infinity.
• Practice Regularly: Consistent practice is key to mastering these concepts. Work through numerous problems of varying difficulty.
• Seek Help When Needed: Don't hesitate to ask your instructor, teaching assistant, or classmates for help if you're struggling with a particular problem.
• Utilize Online Resources: Many online resources, including videos and practice problems, can supplement your learning.
• Understand the Underlying Concepts: Don't just memorize formulas; strive to understand the underlying concepts and how the different techniques work.

By diligently following these steps and consistently practicing, you'll develop the skills and confidence necessary to tackle any infinite limits and limits at infinity homework problems you encounter. Remember, mastering calculus is a journey, and each problem solved brings you closer to a deeper understanding. Good luck!