1.7 Infinite Limits And Limits At Infinity Homework

1.7 Infinite Limits and Limits at Infinity: Homework Demystified

Calculus often presents challenges, and the concept of infinite limits and limits at infinity can be particularly tricky for students. This comprehensive guide will break down the core concepts, provide practical examples, and offer strategies for tackling homework problems related to Section 1.7 (or a similar section covering this topic) in your calculus textbook. We'll delve into the intricacies of these limits, exploring both the theoretical underpinnings and the practical applications.

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 (either positively or negatively). We denote this using the symbols ∞ (infinity) and -∞ (negative infinity). It's crucial to remember that infinity is not a number; it represents unbounded growth.

Key characteristics of infinite limits:

• Vertical Asymptotes: Infinite limits often occur at vertical asymptotes. A vertical asymptote is a vertical line (x = c) where the function approaches infinity or negative infinity as x approaches c.
• One-sided Limits: It's important to consider both left-hand and right-hand limits. A function may approach positive infinity from one side and negative infinity from the other.
• Notation: We write lim_x→c⁺ f(x) = ∞ to indicate that the limit of f(x) as x approaches c from the right is positive infinity. Similar notations exist for other combinations (e.g., lim_x→c^- f(x) = -∞).

Example: Consider the function f(x) = 1/x. As x approaches 0 from the right (x → 0⁺), f(x) approaches positive infinity. As x approaches 0 from the left (x → 0^-), f(x) approaches negative infinity. Therefore, we have:

lim_x→0⁺ 1/x = ∞ and lim_x→0^- 1/x = -∞

This indicates a vertical asymptote at x = 0.

Understanding Limits at Infinity

A limit at infinity describes the behavior of a function as its input grows without bound (x → ∞ or x → -∞). This helps determine if the function approaches a horizontal asymptote.

Key characteristics of limits at infinity:

• Horizontal Asymptotes: Limits at infinity often identify horizontal asymptotes. A horizontal asymptote is a horizontal line (y = L) that the function approaches as x approaches infinity or negative infinity.
• End Behavior: Limits at infinity describe the end behavior of a function—what happens to the function's values as x becomes extremely large (positive or negative).
• Rational Functions: For rational functions (ratios of polynomials), the limit at infinity can often be determined by examining the degrees of the numerator and denominator polynomials.

Example: Consider the function g(x) = (2x² + 1) / (x² - 3). As x approaches infinity, the highest-power terms dominate. Therefore, we can simplify the expression:

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

This signifies a horizontal asymptote at y = 2.

Solving Homework Problems: A Step-by-Step Approach

Let's tackle typical homework problems involving infinite limits and limits at infinity. These problems often involve rational functions, trigonometric functions, or combinations thereof.

Problem 1: Finding Infinite Limits

Find the limits:

a) lim_x→2⁺ (x-3)/(x-2) b) lim_x→2^- (x-3)/(x-2) c) lim_x→2 (x-3)/(x-2)

Solution:

a) As x approaches 2 from the right (x → 2⁺), (x-3) approaches -1, and (x-2) approaches 0 from the positive side. Therefore, the limit is -∞.

b) As x approaches 2 from the left (x → 2^-), (x-3) approaches -1, and (x-2) approaches 0 from the negative side. Therefore, the limit is ∞.

c) Since the left-hand and right-hand limits are not equal, the limit lim_x→2 (x-3)/(x-2) does not exist.

Problem 2: Finding Limits at Infinity for Rational Functions

Find the limit:

lim_x→∞ (3x³ - 2x + 1) / (x³ + 5x² - 7)

Solution:

When dealing with limits at infinity for rational functions, the highest-degree terms in the numerator and denominator dominate. Divide both numerator and denominator by the highest power of x (which is x³ in this case):

lim_x→∞ (3 - 2/x² + 1/x³) / (1 + 5/x - 7/x³)

As x approaches infinity, terms like 2/x², 1/x³, 5/x, and 7/x³ all approach 0. Therefore:

lim_x→∞ (3 - 2/x² + 1/x³) / (1 + 5/x - 7/x³) = 3/1 = 3

The limit is 3, indicating a horizontal asymptote at y = 3.

Problem 3: Limits Involving Trigonometric Functions

Find the limit:

lim_x→∞ (sin(x))/x

Solution:

Recall that the sine function oscillates between -1 and 1. As x approaches infinity, 1/x approaches 0. Therefore, the limit is 0:

lim_x→∞ (sin(x))/x = 0. This implies a horizontal asymptote at y = 0.

Problem 4: A More Complex Example

Evaluate lim_x→∞ [(√(x²+1) - x)]

Solution: This limit requires a bit of algebraic manipulation. We can multiply the expression by the conjugate of the numerator:

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)]

As x approaches infinity, the denominator grows without bound, causing the expression to approach 0. Therefore:

lim_x→∞ [(√(x²+1) - x)] = 0

Strategies for Success

• Master the Definitions: Thoroughly understand the definitions of infinite limits and limits at infinity. This is foundational to solving problems.
• Graphing Calculator: Use a graphing calculator to visualize the behavior of functions. Seeing the graph can help you interpret limits.
• Algebraic Manipulation: Many problems require algebraic simplification (like multiplying by conjugates or factoring) to find the limit.
• L'Hôpital's Rule (Advanced): For indeterminate forms (0/0 or ∞/∞), L'Hôpital's Rule is a powerful tool. However, ensure you understand its application and conditions before using it.
• Practice, Practice, Practice: Work through numerous examples and problems. The more you practice, the more comfortable you’ll become with recognizing patterns and applying the techniques. Start with simpler problems and gradually progress to more challenging ones.

By understanding the concepts, practicing with diverse problems, and employing the right techniques, you can confidently tackle your homework assignments on infinite limits and limits at infinity. Remember that persistent effort is key to mastering these concepts in calculus.