1.7 Infinite Limits and Limits at Infinity Homework Answer Key – This article provides a comprehensive, step‑by‑step guide to solving the typical exercises found in Section 1.7 of a first‑semester calculus textbook. It explains the underlying concepts, walks through sample problems, highlights common pitfalls, and supplies a ready‑to‑use answer key that can be referenced while completing homework assignments.
Introduction
When students first encounter infinite limits and limits at infinity, they often feel overwhelmed by the abstract notation and the idea that a function can “grow without bound” or “approach a horizontal line” as the input becomes arbitrarily large. This section of the textbook (often labeled 1.7) consolidates these ideas into a set of procedural rules that, once mastered, allow learners to evaluate limits quickly and confidently. The purpose of this piece is to transform those procedural rules into an actionable answer key, so you can verify your solutions, understand the reasoning behind each step, and gain the confidence needed to tackle any problem of this type.
Understanding Infinite Limits
What Is an Infinite Limit?
An infinite limit occurs when the function values increase or decrease without bound as the variable approaches a certain point. In notation, we write:
- (\displaystyle \lim_{x \to a} f(x) = \infty) if (f(x)) grows arbitrarily large.
- (\displaystyle \lim_{x \to a} f(x) = -\infty) if (f(x)) decreases without bound.
The key idea is that the limit does not exist as a finite number; instead, it describes the behavior of the function near the point (a).
How to Identify an Infinite Limit
- Examine the denominator – If a rational function has a denominator that approaches zero while the numerator approaches a non‑zero value, the fraction will blow up.
- Check the sign – Determine whether the approaching zero in the denominator is from the positive or negative side; this decides whether the limit is (+\infty) or (-\infty).
- Consider one‑sided behavior – Sometimes the left‑hand and right‑hand limits differ; note these distinctions.
Limits at Infinity
DefinitionA limit at infinity looks at the behavior of a function as the variable heads toward positive or negative infinity. Formally:
- (\displaystyle \lim_{x \to \infty} f(x) = L) if (f(x)) gets arbitrarily close to (L) as (x) becomes large.
- (\displaystyle \lim_{x \to -\infty} f(x) = L) if the same holds as (x) becomes large negative.
When (L) is a finite number, we say the function approaches that horizontal line. When (L) is infinite, we are back to the infinite limit scenario, but now the “approach” is as (x) itself moves without bound.
Typical Cases
| Function Type | Typical Limit at (\infty) | Typical Limit at (-\infty) |
|---|---|---|
| Polynomial of degree (n) with leading coefficient (a) | (\displaystyle \begin{cases}a x^{n} \to \infty & a>0, n\text{ even}\ a x^{n} \to -\infty & a<0, n\text{ odd}\end{cases}) | Same pattern, sign may flip depending on parity |
| Rational function (\frac{p(x)}{q(x)}) | Ratio of leading terms | Same as above, but consider sign of denominator |
| Exponential (a^{x}) (with (a>1)) | (\infty) | (0) (if (0<a<1)) |
| Trigonometric (\sin x, \cos x) | does not exist (oscillates) | does not exist |
Step‑by‑Step Solutions
Below are three representative problems that frequently appear in the 1.7 worksheet, followed by detailed solutions and the final answer key.
Problem 1
Evaluate (\displaystyle \lim_{x \to 2^{+}} \frac{3}{x-2}).
Solution
- As (x) approaches (2) from the right, the denominator (x-2) is a tiny positive number.
- The numerator stays constant at (3).
- A positive constant divided by an increasingly small positive number yields a value that grows without bound.
- That's why, the limit is (+\infty).
Answer: (\boxed{+\infty})
Problem 2
Find (\displaystyle \lim_{x \to \infty} \frac{5x^{2} - 3x + 1}{2x^{2} + 7}).
Solution
- Identify the highest power of (x) present in both numerator and denominator; here it is (x^{2}).
- Divide every term by (x^{2}):
[ \frac{5 - \frac{3}{x} + \frac{1}{x^{2}}}{2 + \frac{7}{x^{2}}} ] - As (x \to \infty), the fractions (\frac{3}{x}), (\frac{1}{x^{2}}), and (\frac{7}{x^{2}}) all tend to (0).
- The expression simplifies to (\frac{5}{2}).
- Hence, the limit is the finite number (\frac{5}{2}).
Answer: (\boxed{\frac{5}{2}})
Problem 3
Determine (\displaystyle \lim_{x \to -\infty} \frac{-4x^{3}+2x}{x^{3}+5}).
Solution
- The dominant term in both numerator and denominator is the cubic term.
- Factor (x^{3}) out of each polynomial:
[ \frac{-4 + \frac{2}{x^{2}}}{1 + \frac{5}{x^{3}}} ] - As (x \to -\infty), the fractions (\frac{2}{x^{2}}) and (\frac{5}{x^{3}}) approach (0).
- The limit reduces to (-4).
- Because the highest power is odd and the leading coefficient is negative, the sign of the limit matches the sign of the coefficient, which is already captured by (-4).
Answer: (\boxed{-4})
Common Mistakes and How to Avoid Them
- Confusing one‑sided limits with two‑sided limits. Always check whether the approach is from the left, right, or both.
- Dropping the sign when dividing by a negative denominator. The sign determines whether the infinite limit is (+\infty) or (-\infty).
- Assuming that any rational function with a zero denominator yields an infinite limit. If the numerator also approaches zero, you may
Common Mistakes and How to Avoid Them (continued)
- Assuming that any rational function with a zero denominator yields an infinite limit. If the numerator also approaches zero, the expression is an indeterminate form (0/0). In such cases you must either factor and cancel common terms, apply L’Hôpital’s Rule, or use series expansions to uncover the true behavior.
- Treating oscillatory functions as if they have a limit. Functions like (\sin x) and (\cos x) do not settle to a single value as (x\to\pm\infty); their limits do not exist. On the flip side, when they appear in a product with a factor that tends to zero (e.g., ( \frac{\sin x}{x})), the whole expression can have a finite limit.
- Ignoring the effect of the sign of the leading coefficient when the highest powers are odd. For odd‑degree polynomials the sign of the coefficient determines whether the function heads to (+\infty) or (-\infty) as (x\to\pm\infty).
Additional Practice Problems
| # | Limit Expression | Expected Difficulty |
|---|---|---|
| 4 | (\displaystyle \lim_{x\to 0^{+}} \frac{1}{\sqrt{x}}) | Easy |
| 5 | (\displaystyle \lim_{x\to \infty} \frac{e^{2x}}{x^{5}}) | Medium |
| 6 | (\displaystyle \lim_{x\to \pi/2^{-}} \tan x) | Easy |
| 7 | (\displaystyle \lim_{x\to -\infty} \frac{7x^{4}+3x^{2}}{-2x^{4}+x}) | Medium |
| 8 | (\displaystyle \lim_{x\to 0} \frac{\sin(5x)}{x}) | Easy (use the standard (\sin x/x) limit) |
| 9 | (\displaystyle \lim_{x\to \infty} \frac{\ln x}{x}) | Medium |
| 10 | (\displaystyle \lim_{x\to 0^{+}} x\ln x) | Hard (requires rewriting as (\frac{\ln x}{1/x}) and applying L’Hôpital) |
Solutions are provided at the end of the worksheet for self‑checking.
Tips for Mastering Limits at Infinity
- Identify the dominant term in the numerator and denominator. The term with the highest power of (x) (or the fastest‑growing exponential) dictates the limit’s behavior.
- Factor out the dominant term to simplify the expression. After factoring, the remaining fractions usually tend to zero, leaving a constant ratio of the leading coefficients.
- Remember the special cases:
- If the degree of the numerator is larger than that of the denominator, the limit is (\pm\infty) (sign follows the leading coefficients).
- If the degree of the denominator exceeds that of the numerator, the limit is (0).
- Equal degrees give the ratio of the leading coefficients.
- Use L’Hôpital’s Rule only when you have a genuine indeterminate form ((0/0) or (\infty/\infty)). Verify the hypothesis first!
- Check one‑sided limits when the function is undefined at a point (e.g., vertical asymptotes). The sign of the denominator’s approach determines whether you get (+\infty) or (-\infty).
- Don’t forget about non‑polynomial functions: exponentials dominate any polynomial, while logarithms are dominated by any power of (x). Trigonometric functions oscillate, so they only yield a limit when multiplied by a term that forces the product to zero.
Answer Key for the Worksheet
| # | Limit | Result |
|---|---|---|
| 1 | (\displaystyle \lim_{x \to 2^{+}} \frac{3}{x-2}) | (+\infty) |
| 2 | (\displaystyle \lim_{x \to \infty} \frac{5x^{2} - 3x + 1}{2x^{2} + 7}) | (\dfrac{5}{2}) |
| 3 | (\displaystyle \lim_{x \to -\infty} \frac{-4x^{3}+2x}{x^{3}+5}) | (-4) |
| 4 | (\displaystyle \lim_{x\to 0^{+}} \frac{1}{\sqrt{x}}) | (+\infty) |
| 5 | (\displaystyle \lim_{x\to \infty} \frac{e^{2x}}{x^{5}}) | (+\infty) |
| 6 | (\displaystyle \lim_{x\to \pi/2^{-}} \tan x) | (+\infty) |
| 7 | (\displaystyle \lim_{x\to -\infty} \frac{7x^{4}+3x^{2}}{-2x^{4}+x}) | (-\dfrac{7}{2}) |
| 8 | (\displaystyle \lim_{x\to 0} \frac{\sin(5x)}{x}) | (5) |
| 9 | (\displaystyle \lim_{x\to \infty} \frac{\ln x}{x}) | (0) |
| 10 | (\displaystyle \lim_{x\to 0^{+}} x\ln x) | (0) |
Short version: it depends. Long version — keep reading.
Conclusion
Understanding limits as (x) heads toward infinity (or toward a point where the function blows up) is a cornerstone of calculus. By focusing on the dominant terms, carefully tracking signs, and applying the rules for polynomial degrees, you can quickly determine whether a limit is a finite number, zero, or an infinite divergence.
It sounds simple, but the gap is usually here.
Remember:
- Higher‑order growth beats lower‑order growth (exponential (\gg) polynomial (\gg) logarithmic).
- One‑sided limits reveal the direction of divergence near vertical asymptotes.
- Oscillatory functions only have limits when combined with a factor that forces the product to settle.
With the strategies, examples, and practice problems presented here, you should now feel equipped to tackle any limit‑at‑infinity question that appears on the 1.Keep practicing, watch for the subtle sign changes, and the patterns will become second nature. Day to day, 7 worksheet—or on a future exam. Happy calculating!
