1.7 Infinite Limits And Limits At Infinity Homework Answer Key

1.7 Infinite Limits and Limits at Infinity: Homework Answer Key & Comprehensive Guide

This comprehensive guide delves into the intricacies of infinite limits and limits at infinity, crucial concepts within the realm of calculus. We'll explore these concepts theoretically and practically, providing a detailed explanation and working through numerous examples to solidify your understanding. This guide acts as a virtual homework answer key, but more importantly, it aims to equip you with the skills and knowledge to tackle similar problems independently. We'll cover various techniques and scenarios, ensuring you're well-prepared to handle any challenges related to infinite limits and limits at infinity.

Understanding Infinite Limits

An infinite limit describes the behavior of a function as its input approaches a specific value, resulting in an output that grows without bound (approaches positive or negative infinity). This is denoted as:

• lim_x→a f(x) = ∞: The limit of f(x) as x approaches 'a' is positive infinity.
• lim_x→a f(x) = -∞: The limit of f(x) as x approaches 'a' is negative infinity.

Key characteristics of infinite limits:

• Vertical Asymptotes: Infinite limits often indicate the presence of a vertical asymptote at x = a. The function's graph approaches infinity or negative infinity as x gets closer to 'a'.
• One-sided Limits: It's crucial to examine both the left-hand limit (lim_x→a^- f(x)) and the right-hand limit (lim_x→a⁺ f(x)) to determine the behavior of the function near the asymptote. They may approach different infinities or even be finite.
• Discontinuities: Infinite limits signify a type of discontinuity – an infinite discontinuity – where the function's value is undefined at x = a.

Examples of Infinite Limits

Let's work through some examples to illustrate the concept:

Example 1: Find lim_x→0 (1/x)

As x approaches 0 from the positive side (x → 0⁺), 1/x becomes increasingly large, approaching positive infinity. As x approaches 0 from the negative side (x → 0^-), 1/x becomes increasingly large in the negative direction, approaching negative infinity.

Therefore:

• lim_x→0⁺ (1/x) = ∞
• lim_x→0^- (1/x) = -∞

The limit lim_x→0 (1/x) does not exist because the left-hand and right-hand limits are different.

Example 2: Find lim_x→2 (1/(x-2)²)

As x approaches 2 from either side, (x-2)² approaches 0, but always remains positive. Consequently, 1/(x-2)² becomes arbitrarily large, tending towards positive infinity.

Therefore: lim_x→2 (1/(x-2)²) = ∞

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

Notice that the numerator can be factored as (x+1)². Thus, the expression simplifies to: (x+1)²/(x+1) = x+1 for x ≠ -1. Taking the limit as x approaches -1, we get:

lim_x→-1 (x+1) = 0. Therefore, the limit exists and is equal to 0, despite the initial expression appearing undefined at x = -1.

Understanding Limits at Infinity

A limit at infinity describes the behavior of a function as its input grows without bound (approaches positive or negative infinity). This indicates the function's eventual behavior, its horizontal asymptotes, if any. It's represented as:

• lim_x→∞ f(x) = L: The limit of f(x) as x approaches infinity is L.
• lim_x→-∞ f(x) = L: The limit of f(x) as x approaches negative infinity is L.

Key Characteristics of Limits at Infinity:

• Horizontal Asymptotes: Limits at infinity often identify horizontal asymptotes. If the limit is L, then y = L is a horizontal asymptote. A function can have at most two horizontal asymptotes, one for x→∞ and another for x→-∞.
• End Behavior: Limits at infinity describe the function's end behavior – how it behaves as x extends to very large positive or negative values.
• Rational Functions: Evaluating limits at infinity for rational functions often involves dividing both the numerator and denominator by the highest power of x.

Examples of Limits at Infinity

Let's explore some examples to understand limits at infinity:

Example 1: Find lim_x→∞ (1/x)

As x approaches infinity, 1/x approaches 0.

Therefore: lim_x→∞ (1/x) = 0

Example 2: Find lim_x→-∞ (1/x)

Similarly, as x approaches negative infinity, 1/x approaches 0.

Therefore: lim_x→-∞ (1/x) = 0

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

To evaluate this limit, we divide both the numerator and the denominator by the highest power of x, which is x²:

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

As x approaches infinity, terms like 2/x, 1/x², 5/x, and 2/x² all approach 0. Therefore, the limit simplifies to:

lim_x→∞ (3 + 0 - 0) / (1 - 0 + 0) = 3/1 = 3

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

Example 4: Find lim_x→∞ (x³ + 2x) / (x² - 1)

Dividing both the numerator and denominator by x² (the highest power in the denominator):

lim_x→∞ (x + 2/x) / (1 - 1/x²)

As x approaches infinity, the term x becomes unbounded, leading to an infinite limit.

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

Techniques for Evaluating Limits

Several techniques are employed to evaluate limits at infinity and infinite limits:

• Algebraic Manipulation: Factoring, simplifying expressions, and canceling common factors are crucial for simplifying complex expressions before evaluating limits.
• L'Hôpital's Rule: This powerful rule applies to indeterminate forms (0/0 or ∞/∞). If the limit is indeterminate, differentiate the numerator and denominator separately and then evaluate the limit of the resulting expression. This process can be repeated as needed.
• Squeeze Theorem: If a function is bounded between two other functions that converge to the same limit, then the function also converges to that limit.
• Substitution: Sometimes, a substitution can simplify the expression, making it easier to evaluate the limit.

Homework Problems and Solutions (Illustrative Examples)

This section provides a series of homework-style problems with detailed solutions to further solidify your understanding. These examples cover various scenarios and techniques discussed previously.

Problem 1: Evaluate lim_x→3 (x² - 9) / (x - 3)

Solution: Factor the numerator: (x - 3)(x + 3). Cancel the common factor (x - 3).

The expression simplifies to (x + 3) for x ≠ 3. Now, substitute x = 3:

lim_x→3 (x + 3) = 3 + 3 = 6

Problem 2: Evaluate lim_x→∞ (4x³ - 2x + 1) / (2x³ + x² - 5)

Solution: Divide both the numerator and denominator by x³:

lim_x→∞ (4 - 2/x² + 1/x³) / (2 + 1/x - 5/x³)

As x → ∞, the terms with x in the denominator approach 0. Therefore, the limit simplifies to 4/2 = 2.

Problem 3: Evaluate lim_x→0 (sin x) / x

Solution: This is a standard limit that's often encountered. The limit is 1.

Problem 4: Evaluate lim_x→∞ (e^x)

Solution: As x approaches infinity, e^x grows without bound. Therefore, the limit is ∞.

Problem 5: Evaluate lim_x→-∞ (e^x)

Solution: As x approaches negative infinity, e^x approaches 0. Therefore, the limit is 0.

Problem 6: Evaluate lim_x→∞ (ln x)

Solution: As x approaches infinity, ln x also grows without bound. Therefore, the limit is ∞.

Problem 7: Using L'Hôpital's Rule, Evaluate lim_x→0 (sin x) / x

Solution: The initial expression is in the indeterminate form 0/0. Applying L'Hôpital's rule (differentiating the numerator and denominator), we get:

lim_x→0 (cos x) / 1 = cos(0) = 1

Problem 8: Evaluate lim_x→∞ (x² + 1) / (x³ - 1)

Solution: Divide by the highest power of x in the denominator (x³):

lim_x→∞ (1/x + 1/x³) / (1 - 1/x³) = 0/1 = 0

These problems demonstrate the application of various techniques to solve diverse limit problems. Remember to carefully analyze the problem, identify the applicable technique, and meticulously apply the steps to arrive at the correct solution. Consistent practice is key to mastering these concepts.

Conclusion

Understanding infinite limits and limits at infinity is essential for mastering calculus. This guide provides a comprehensive overview of these concepts, including definitions, characteristics, and various techniques for evaluating limits. The numerous examples and solved homework problems serve as a strong foundation for further exploration and application of these crucial concepts in more advanced calculus topics. Remember that consistent practice is paramount to developing a strong understanding and proficiency in evaluating limits.