← Back to 5. The derivative in Graphing and Applications

📝 Oblique slant asymptotes rational functions (19 MCQs)

📖 From Calculus • 5. The derivative in Graphing and Applications • 19 questions available

←Previous Topic:Graphing rational functions step by step📝 Oblique slant asymptotes rational functionsVertical tangents and cusps in graphs:Next Topic→

What is Oblique slant asymptotes rational functions?

Definition:
An oblique or slant asymptote occurs when the degree of the numerator is exactly one greater than the denominator. It is found by polynomial long division, where the quotient (ignoring remainder) gives the linear equation $y = mx + b$ approached by the graph at infinity.

Example:
For $f(x) = \frac{x^2+1}{x}$ , division yields $x + \frac{1}{x}$ . The slant asymptote is $y = x$ , as $\frac{1}{x} \to 0$ when $x \to \infty$ .

Reason:
Slant asymptotes describe the end behavior of improper rational functions, showing how the graph aligns with a non-horizontal line as inputs become very large or small.

Easy

Medium

Hard

📝 All Oblique slant asymptotes rational functions MCQs

Q1. For the rational function $f(x)=\frac{2x^{2}+3x-5}{x-1}$ , what is its oblique (slant) asymptote?

y=2x+5

✅

y=2x+3

y=2x+1

y=2x-1

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Perform polynomial long division of the numerator by the denominator. Dividing $2x^{2}+3x-5$ by $x-1$ yields a quotient of $2x+5$ with zero remainder, so the graph approaches the line $y=2x+5$ as $x\to\pm\infty$ . Hence that line is the slant asymptote.

Q2. Which condition guarantees that a rational function $f(x)=\frac{p(x)}{q(x)}$ has an oblique asymptote?

\deg p > \deg q +1

\deg p = \deg q

\deg p = \deg q +1

✅

\deg p < \deg q

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: An oblique (slant) asymptote occurs precisely when the degree of the numerator exceeds the degree of the denominator by exactly one. In that case polynomial long division produces a linear quotient, which is the line that the function approaches for large $|x|$ .

Q3. Consider $f(x)=\frac{x^{2}-4}{x-2}$ . Which statement correctly describes its vertical feature and its slant asymptote?

A.Vertical asymptote at

x=2

, oblique asymptote

y=x+2

B.Hole at

x=2

, oblique asymptote \(y=x+2\ ✅

C.Vertical asymptote at

x=2

, no oblique asymptote

D.Hole at

x=2

, no asymptote

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Factor the numerator as $(x-2)(x+2)$ and cancel the common factor with the denominator. The resulting function is $x+2$ with a removable discontinuity (hole) at $x=2$ . The line $y=x+2$ is the slant asymptote because the original rational function approaches that line as $|x|$ grows.

Q4. What is the definition of an oblique (slant) asymptote for a rational function?

A.A horizontal line y = c

B.A vertical line x = a

C.A linear function y = mx + b where the numerator degree exceeds the denominator degree by one ✅

D.A parabola y = ax^2 + bx + c

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: An oblique asymptote occurs when the numerator’s degree is exactly one higher than the denominator’s. Polynomial long division then yields a linear term y = mx + b, which the graph approaches as |x| → ∞, distinguishing it from horizontal or vertical asymptotes.

Q5. Which method is commonly used to find the equation of an oblique asymptote of a rational function?

A.Synthetic division

B.Polynomial long division ✅

C.Factoring the numerator

D.Completing the square

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Polynomial long division is the standard technique because it directly produces the quotient, which is the linear expression representing the oblique asymptote, and the remainder term that indicates how the function deviates from that line for large |x|.

Q6. Given f(x)= $\frac{2x^{2}+3x+1}{x-1}$ and its oblique asymptote $y=2x+5$ , for which interval does $f(x)$ lie above the asymptote?

x<1

x>1

✅

C.all

x\neq1

D.NONE

💡 Difficulty: medium | ✅ Correct: B

Q7. If a rational function has an oblique asymptote $y=mx+b$ , what is $\displaystyle\lim_{x\to\infty}\bigl(f(x)-(mx+b)\bigr)$ ?

A.0 ✅

B.∞

C.m

D.DOES NOT EXIST

💡 Difficulty: easy | ✅ Correct: A

Q8. Given $f(x)=\frac{2x^{2}+3x-5}{x-1}$ , what is its oblique asymptote?

y=2x+5

✅

y=2x+3

y=2x+1

y=2x-1

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Perform polynomial division: $2x^{2}+3x-5$ divided by $x-1$ yields quotient $2x+5$ with zero remainder, so the slant (oblique) asymptote is $y=2x+5$ . This matches option A.

Q9. An oblique asymptote of a rational function is a:

A.Horizontal line

B.Vertical line

C.Slant (non‑horizontal, non‑vertical) line ✅

D.Curved line

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: By definition, an oblique asymptote is a straight line that is neither horizontal nor vertical, often called a slant asymptote. It occurs when the numerator’s degree exceeds the denominator’s by exactly one.

Q10. For $f(x)=\frac{x^{2}-4}{x-2}$ , what type of discontinuity occurs at $x=2$ ?

A.Hole (removable discontinuity)

B.Vertical asymptote

C.Oblique asymptote

D.Jump discontinuity ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Factor the numerator: $(x-2)(x+2)$ . Cancelling the common factor leaves $f(x)=x+2$ except at $x=2$ , where the function is undefined, creating a removable discontinuity (hole).

Q11. Find the x‑intercept(s) of $f(x)=\frac{x^{2}-9}{x+3}$ .

x=3

only ✅

x=-3

only

C.Both

x=3

and

x=-3

D.No x‑intercepts

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Set the numerator zero: $x^{2}-9=0$ gives $x=\pm3$ . However, $x=-3$ also zeros the denominator, producing a hole, not an intercept. Thus the only valid x‑intercept is at $x=3$ .

Q12. If a rational function has an oblique asymptote $y=mx+b$ , which relationship must hold between the degrees of numerator and denominator?

\deg\text{numerator}=\deg\text{denominator}+1

✅

\deg\text{numerator}=\deg\text{denominator}

\deg\text{numerator}=\deg\text{denominator}+2

D.Denominator degree larger than numerator degree

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: A slant (oblique) asymptote appears precisely when the numerator’s degree exceeds the denominator’s by one, giving a linear quotient after division.

Q13. Determine the oblique asymptote of $f(x)=\frac{x^{3}+2x^{2}-3x+1}{x^{2}+1}$ .

y=x+2

✅

y=x+1

y=x-1

y=x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Dividing the cubic by the quadratic gives quotient $x+2$ with a remainder. The linear part $y=x+2$ is the slant asymptote.

Q14. Find the slant asymptote of $f(x)=\frac{3x^{2}+4x+1}{x-2}$ .

y=3x+10

✅

y=3x+8

y=3x+12

y=3x+9

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Long division of $3x^{2}+4x+1$ by $x-2$ yields quotient $3x+10$ and a constant remainder, so the oblique asymptote is $y=3x+10$ .

Q15. For $f(x)=\frac{x^{2}+1}{x-1}$ , on which interval is the function decreasing?

(-\infty,0)

(0,1)

✅

(1,\infty)

(-\infty,1)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Rewrite $f=x+1+\frac{2}{x-1}$ . Its derivative is $1-\frac{2}{(x-1)^{2}}$ . Setting $f'<0$ gives $(x-1)^{2}<2$ , i.e., $1-\sqrt2<x<1+\sqrt2$ . The sub‑interval $(0,1)$ lies inside this region, so the function decreases there.

Q16. Compare the oblique asymptotes of $f(x)=\frac{2x^{3}+3x^{2}-5x+1}{x^{2}+1}$ and $g(x)=\frac{2x^{3}-4x+2}{x^{2}-1}$ . Which statement is true?

A.Both have the same slope ✅

f

has a larger intercept

g

has a vertical asymptote at

x=1

f

's asymptote is

y=2x+3

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Dividing each numerator by its denominator gives linear quotients $2x+3$ for $f$ and $2x-2$ for $g$ . Both share slope $2$ ; thus the statement about equal slopes is correct.

Q17. For $h(x)=\frac{x^{3}-x}{x^{2}-4}$ , find its oblique asymptote and state whether the graph approaches it from above or below as $x\to\infty$ .

y=x

, from above ✅

y=x

, from below

y=x+1

, from above

y=x+1

, from below

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Long division yields quotient $x$ and remainder $3x$ . The remainder term $3x/(x^{2}-4)$ is positive for large $x$ , so $h(x)=x+3/x+o(1)$ approaches $y=x$ from above.

Q18. If $f(x)=\frac{ax^{2}+bx+c}{dx+e}$ has an oblique asymptote $y=mx+n$ , what is $m$ in terms of $a$ and $d$ ?

a/d

✅

d/a

a

d

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: When dividing the quadratic numerator by the linear denominator, the leading term of the quotient is $(a/d)x$ . Hence the slope of the slant asymptote is $m=a/d$ .

Q19. Evaluate $\displaystyle\lim_{x\to\infty}\Bigl[\frac{5x^{2}+3x+1}{x+2}-(5x-7)\Bigr]$ .

A.0 ✅

B.1

C.-1

D.2

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Perform division: $\frac{5x^{2}+3x+1}{x+2}=5x-7+\frac{15}{x+2}$ . Subtracting $5x-7$ leaves $\frac{15}{x+2}$ , whose limit as $x\to\infty$ is $0$ .

🔗 Related Topics (MCQs)

📝 Increasing and decreasing functions using derivative MCQs 📝 Concavity and inflection points calculus MCQs 📝 Logistic growth curves calculus MCQs 📝 Analysis of Functions II (Relative Extrema) in calculus MCQs 📝 Relative maxima and minima local extrema MCQs

←Previous Topic:Graphing rational functions step by step📝 Oblique slant asymptotes rational functionsVertical tangents and cusps in graphs:Next Topic→