π Limits at infinity as x approaches infinity (18 MCQs)
π From Calculus β’ 2. Limits and Continuity an Introduction β’ 18 questions available
What is Limits at infinity as x approaches infinity?
Definition:
A limit at infinity, denoted , describes the value that approaches as increases without bound (becomes arbitrarily large). This captures the long-term behavior or end behavior of the function, indicating whether it stabilizes to a horizontal asymptote, grows without bound, or oscillates, and is essential for understanding function behavior for large inputs.
Example:
Evaluate .
Solution: As , , so the limit is .
Reason:
This concept is fundamental for analyzing asymptotic behavior, which is crucial in fields like economics for long-term trends, physics for terminal velocity, and engineering for system stability at large scales.
π All Limits at infinity as x approaches infinity MCQs
Q1. What is the definition of a horizontal asymptote for a function f(x) as xβ+β?
π Explanation: A horizontal asymptote is a horizontal line that the graph of the function approaches as x grows without bound. Formally, if the limit of f(x) as xβ+β equals a constant L, then y = L is a horizontal asymptote.
Q2. Evaluate lim_{xβ+β} 1/x.
π Explanation: As x becomes larger and larger, the denominator grows without bound while the numerator stays at 1. The fraction therefore gets arbitrarily close to zero, so the limit is 0. The answer choice representing zero is the correct one.
Q3. Suppose lim_{xβ+β} f(x)=0. Which statement must be true about lim_{xβ+β} 1/(f(x)+1)?
π Explanation: If f(x) approaches 0, then f(x)+1 approaches 1. The reciprocal of a quantity that approaches 1 also approaches 1. Therefore the limit of 1/(f(x)+1) is 1, making the fourth option the correct choice.
Q4. If lim_{xβ-β} g(x)=5 and h(x)=2g(x)-3, what is lim_{xβ-β} h(x)?
π Explanation: Substituting the limiting value of g(x) into h(x) gives h(x)β2Β·5β3=10β3=7. Hence the limit of h(x) as xβ-β is 7, which appears as the second option.
Q5. Given that lim_{xβ+β} p(x)=L exists and is finite, which statement about lim_{xβ+β} p(x)/x is necessarily true?
π Explanation: Since p(x) approaches a finite number L, the numerator stays bounded while the denominator grows without bound. A bounded quantity divided by an unbounded one tends to 0, so the limit must be 0.
Q6. For the function f(x)=(x^2-1)/(x^2+2x+1), determine lim_{xβ-β} f(x).
π Explanation: Both numerator and denominator are quadratic with leading coefficient 1. As |x| becomes large, lowerβorder terms become negligible, so the ratio approaches 1. The third option correctly states this limit.
Q7. Compare the limits lim_{xβ+β}(2x+3)/x and lim_{xβ+β} x/(2x+3). Which statement is correct?
π Explanation: The first expression simplifies to 2+3/x, whose limit is 2. The second simplifies to 1/(2+3/x), whose limit is 0 because the denominator tends to 2, making the fraction approach 0. Hence the second option is correct.
Q8. Which rational function has a horizontal asymptote at y=3 as xβ+β?
π Explanation: The horizontal asymptote is given by the ratio of leading coefficients. For (3x^2+1)/(x^2-4) the ratio is 3/1=3, but the correct answer is placed in option D after rearranging the list, so D is the right choice.
Q9. Consider f(x)=(x^3+2x)/(x^3-5). Which description best fits its end behavior?
π Explanation: Both numerator and denominator have the same highest degree with leading coefficient 1, so the limit is 1. For large positive x the extra +2x makes the numerator slightly larger, so the graph approaches y=1 from above, matching the third option.
Q10. The function f(x)=tan^{-1}x has horizontal asymptotes at which y-values?
π Explanation: As xβ+β, arctanβ―x approaches Ο/2, and as xβ-β it approaches -Ο/2. Thus the graph has two horizontal asymptotes at y=Ο/2 and y=-Ο/2, which is the first answer choice.
Q11. If g(x)=1/x and h(x)=sin(g(x)), what is lim_{xβ+β} h(x)?
π Explanation: When x grows, g(x)=1/x tends to 0. The sine of a quantity that approaches 0 also approaches 0. Therefore the limit of h(x) is 0, which appears as the fourth option.
Q12. For the composite function F(x)=1/atan(x), determine lim_{xβ+β} F(x).
π Explanation: As xβ+β, atanβ―x approaches Ο/2. Taking the reciprocal gives 1/(Ο/2)=2/Ο. Hence the second option correctly gives the limit.
Q13. Let f(x)=x^2/β(x^4+1). Evaluate lim_{xβ-β} f(x).
π Explanation: For large |x|, β(x^4+1)=|x^2|Β·β(1+1/x^4)β|x^2|. Since |x^2|=x^2, the fraction simplifies to x^2/x^2=1. Thus the limit is 1, which is the third option.
Q14. If lim_{xβ-β} k(x)=L and Lβ 0, what is lim_{xβ-β} 1/k(x)?
π Explanation: When k(x) approaches a nonβzero constant L, its reciprocal approaches 1/L. This follows directly from the continuity of the reciprocal function at nonβzero points, making the first option correct.
Q15. For which values of a does f(x)=(x^2+ax+1)/(x^2+1) have a horizontal asymptote y=1 as xβΒ±β?
π Explanation: The horizontal asymptote depends on the ratio of leading coefficients, which are both 1 regardless of a. Therefore the function always approaches y=1 for any real a, so the correct answer is the first option, placed as D after shuffling.
Q16. Evaluate lim_{xβ+β} (e^{x}+x)/e^{x}.
π Explanation: The term e^{x} dominates both the numerator and denominator. Dividing numerator and denominator by e^{x} yields 1 + x/e^{x}, and x/e^{x}β0. Hence the limit equals 1, which is the second option.
Q17. Let f(x)=ln x / x. What is lim_{xβ+β} f(x)?
π Explanation: The logarithm grows much more slowly than the linear term x, so the ratio lnβ―xβ―/β―x tends to 0 as xββ. The fourth option correctly states this limit.
Q18. Suppose lim_{xβ+β} m(x)=5 and lim_{xβ+β} n(x)=β. What is lim_{xβ+β} m(x)/n(x)?
π Explanation: A finite constant divided by a quantity that grows without bound approaches 0. Therefore the limit of the quotient is 0, which corresponds to the first answer choice.