Advanced Limits at Infinity Concepts

Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7682 | Total Attempts: 9,547,133
| Questions: 15 | Updated: Dec 17, 2025
Please wait...
Question 1 / 15
0 %
0/100
Score 0/100
1) Evaluate lim(x→∞) (7x⁵ + 2x - 1)/(3x⁵ - 8x³ + 5)

Explanation

The degrees of numerator and denominator are both 5. Divide numerator and denominator by x⁵: (7 + 2/x^4 - 1/x⁵)/(3 - 8/x² + 5/x⁵). As x → ∞ all terms with x in the denominator go to 0, leaving 7/3.

Submit
Please wait...
About This Quiz
Advanced Limits At Infinity Concepts - Quiz

Here’s where limits at infinity get more interesting! In this quiz, you’ll explore how exponential functions outgrow polynomials, why logarithms increase slowly, and how the squeeze theorem helps evaluate tricky expressions. You’ll analyze limits involving sin(x), ln(x), e^x, and rational expressions where degrees match or differ. By the end, you’ll... see moresee how limits reveal which functions dominate and how to determine horizontal asymptotes in more advanced settings.
see less

2)
You may optionally provide this to label your report, leaderboard, or certificate.
2) Evaluate limx→-∞ (4x³ - 9x)/(5x³ + 2)

Explanation

The degrees of the numerator and denominator are equal (both are 3). For rational functions where the degrees are equal, the limit as x approaches positive or negative infinity is simply the ratio of the leading coefficients. The leading coefficients are 4 and 5, so the limit is 4/5.

Submit
3) Evaluate limx→∞ √(x² + 5x + 1) - x

Explanation

∞ - ∞ form. Multiply by conjugate: [ (x² + 5x + 1) - x² ] / [ √(x² + 5x + 1) + x ] = (5x + 1) / [ √(x² + 5x + 1) + x ]. Divide numerator and denominator by x (x > 0): (5 + 1/x) / [ √(1 + 5/x + 1/x²) + 1 ] → 5 / (1 + 1) = 5/2.

Submit
4) True or False: If the limit as x approaches infinity of f(x) equals a finite number L, then the line y = L is a horizontal asymptote of the graph of f(x).

Explanation

This is the definition of a horizontal asymptote. If the function values f(x) get arbitrarily close to a number L as x increases without bound, then the graph of the function approaches the horizontal line y = L.

Submit
5) Evaluate lim(x→∞) ln(3x + 7) / √x

Explanation

Logarithm grows slower than any positive power of x. √x = x^{1/2}, so ln(something linear) / x^{1/2} → 0.

Submit
6) Evaluate lim(x→-∞) (5x^4 + 3x²)/(-2x^4 + x)

Explanation

Highest degree 4 in both. Leading coefficients 5 and -2, ratio 5/(-2) = -5/2.

Submit
7) Evaluate lim(x→∞) (4 + sin(x))/x

Explanation

|sin(x)| ≤ 1, so |(4 + sin(x))/x| ≤ (4 + 1)/|x| = 5/|x| → 0 as x → ∞. By squeeze theorem the limit is 0.

Submit
8) Evaluate limx→∞ (x + 5)/(x + sin(x))

Explanation

Divide numerator and denominator by x: (1 + 5/x)/(1 + sin(x)/x). sin(x)/x → 0, 5/x → 0, so 1/1 = 1.

Submit
9) Evaluate limx→∞ (2x² + 3x)/(e^x + e^{-x})

Explanation

Denominator ≈ e^x for large positive x, so behaves like (2x²)/e^x → 0.

Submit
10) Evaluate limx→∞ [√(4x² - x) - 2x]

Explanation

Conjugate method: multiply by √(4x² - x) + 2x. Numerator (4x² - x) - 4x² = -x. Denominator x (√(4 - 1/x) + 2). Divide num/den by x: -1 / (√(4 - 1/x) + 2) → -1/(2 + 2) = -1/4.

Submit
11) Evaluate limx→∞ x ( √(1 + 1/x) - 1 )

Explanation

Rationalize: x [ (1 + 1/x - 1) / (√(1 + 1/x) + 1) ] = x (1/x) / (√(1 + 1/x) + 1) = 1 / (√(1 + 1/x) + 1) → 1/(1 + 1) = 1/2.

Submit
12) Evaluate limx→∞ (ln x)⁵ / x

Explanation

Any power of ln x grows slower than any positive power of x. L’Hôpital five times eventually gives constant / x → 0.

Submit
13) Evaluate limx→∞ e^{2x} / (x⁵ + x^100)

Explanation

In the denominator the dominant term is x^100, but exponential still grows faster than any polynomial, so e^{2x} / x^100 → ∞.

Submit
14) Determine the horizontal asymptote of f(x) = (3x² + 5)/(x² - 4x + 7) as x → ∞

Explanation

Same degree, ratio of leading coefficients 3/1 = 3, so horizontal asymptote y = 3.

Submit
15) Evaluate limx→∞ sin(x²)/x

Explanation

Since |sin(anything)| ≤ 1, so |sin(x²)/x| ≤ 1/|x| → 0, so by squeeze theorem the limit is 0.

Submit
×
Saved
Thank you for your feedback!
View My Results
Cancel
  • All
    All (15)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
Evaluate lim(x→∞) (7x⁵ + 2x - 1)/(3x⁵ - 8x³ + 5)
Evaluate limx→-∞ (4x³ - 9x)/(5x³ + 2)
Evaluate limx→∞ √(x² + 5x + 1) - x
True or False: If the limit as x approaches infinity of f(x) equals a...
Evaluate lim(x→∞) ln(3x + 7) / √x
Evaluate lim(x→-∞) (5x^4 + 3x²)/(-2x^4 + x)
Evaluate lim(x→∞) (4 + sin(x))/x
Evaluate limx→∞ (x + 5)/(x + sin(x))
Evaluate limx→∞ (2x² + 3x)/(e^x + e^{-x})
Evaluate limx→∞ [√(4x² - x) - 2x]
Evaluate limx→∞ x ( √(1 + 1/x) - 1 )
Evaluate limx→∞ (ln x)⁵ / x
Evaluate limx→∞ e^{2x} / (x⁵ + x^100)
Determine the horizontal asymptote of f(x) = (3x² + 5)/(x² - 4x + 7)...
Evaluate limx→∞ sin(x²)/x
Alert!

Advertisement