Skip to content

Quiz: Asymptotes and End Behavior

Test your understanding of asymptotes and end behavior with these review questions.

1. What is a vertical asymptote?

  1. A horizontal line that the graph approaches
  2. A vertical line that the graph approaches but never crosses
  3. The highest point on the graph
  4. A line where the function equals zero
Show Answer

The correct answer is B. A vertical asymptote is a vertical line x = a that the graph approaches as the function values become unbounded (approach ±∞). The function is typically undefined at the asymptote.

Concept Tested: Vertical Asymptote

2. Where does f(x) = 1/(x + 2) have a vertical asymptote?

  1. x = 0
  2. x = 2
  3. x = −2
  4. x = 1
Show Answer

The correct answer is C. Vertical asymptotes occur where the denominator equals zero (and the numerator doesn't). Setting x + 2 = 0 gives x = −2.

Concept Tested: Vertical Asymptote

3. What is the horizontal asymptote of f(x) = (3x + 1)/(x − 2)?

  1. y = 0
  2. y = 1
  3. y = 3
  4. No horizontal asymptote
Show Answer

The correct answer is C. When the degrees of numerator and denominator are equal, the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator) = 3/1 = 3.

Concept Tested: Horizontal Asymptote

4. What is lim(x→∞) (2x² + 1)/(x² − 3)?

  1. 0
  2. 2
  3. ∞
  4. −3
Show Answer

The correct answer is B. Both numerator and denominator have degree 2, so divide each term by x²: (2 + 1/x²)/(1 − 3/x²). As x→∞, this approaches 2/1 = 2. This is the horizontal asymptote.

Concept Tested: Limit at Infinity

5. When does a rational function have an oblique (slant) asymptote?

  1. When the degree of the numerator equals the degree of the denominator
  2. When the degree of the numerator is exactly one more than the denominator
  3. When the numerator has a higher degree than the denominator by any amount
  4. When the function has no horizontal asymptote
Show Answer

The correct answer is B. An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. You find it by performing polynomial long division.

Concept Tested: Oblique Asymptote

6. What is the end behavior of f(x) = −2x³?

  1. As x→∞, f(x)→∞; as x→−∞, f(x)→−∞
  2. As x→∞, f(x)→−∞; as x→−∞, f(x)→∞
  3. As x→±∞, f(x)→∞
  4. As x→±∞, f(x)→−∞
Show Answer

The correct answer is B. With odd degree and negative leading coefficient, the ends go in opposite directions: as x→∞, f(x)→−∞ (down right), and as x→−∞, f(x)→∞ (up left).

Concept Tested: End Behavior

7. What is lim(x→2⁺) 1/(x − 2)?

  1. +∞
  2. −∞
  3. 0
  4. 2
Show Answer

The correct answer is A. As x approaches 2 from the right, (x − 2) is small and positive, so 1/(x − 2) is large and positive, approaching +∞. This is a one-sided infinite limit.

Concept Tested: One-Sided Infinite Limit

8. Which function grows faster as x→∞: f(x) = x³ or g(x) = 2ˣ?

  1. f(x) = x³ grows faster
  2. g(x) = 2ˣ grows faster
  3. They grow at the same rate
  4. Neither grows without bound
Show Answer

The correct answer is B. Exponential functions always grow faster than polynomial functions as x→∞. No matter how large the polynomial degree, exponential growth eventually dominates.

Concept Tested: Comparing Growth Rates

9. What is the dominant term of 5x⁴ − 3x³ + 2x − 7 as x→∞?

  1. −7
  2. 2x
  3. −3x³
  4. 5x⁴
Show Answer

The correct answer is D. The dominant term is the term with the highest power of x, which controls the function's behavior as x becomes very large. Here, 5x⁴ dominates all other terms for large x.

Concept Tested: Dominant Term

10. What is lim(x→∞) (5x + 3)/(2x² − 1)?

  1. 5/2
  2. 0
  3. ∞
  4. −3
Show Answer

The correct answer is B. When the degree of the denominator (2) is greater than the degree of the numerator (1), the limit as x→∞ is 0. The horizontal asymptote is y = 0.

Concept Tested: Rational End Behavior