Exercise · ZW-LIMITS-AT-INFINITY-005 · ★★☆ Intermediate
Compute \(\lim_{x\to\infty}\,\frac{\sqrt{x^{2} + 1}}{x}\).
Choose one option.
Show answer
- A. \(1\)
Explanation
For large positive x, \(\sqrt{x^{2}+1}\approx x\), so the quotient approaches 1. Formally, \(\dfrac{\sqrt{x^{2}+1}}{x}=\sqrt{1+\dfrac{1}{x^{2}}}\to1\).
Review the concept: Limits at infinity
The limit of \(f\) as \(x\) tends to infinity describes what happens to \(f(x)\) when \(x\) grows without bound. For a quotient of polynomials, the result is decided by the quotient of the highest-degree terms.
\[\lim_{x\to\infty}\frac{a_n x^{n}+\cdots}{b_n x^{n}+\cdots}=\frac{a_n}{b_n}\]
More practice problems: limits at infinity
- ZW-LIMITS-AT-INFINITY-001 Compute lim _x→∞ (3 x^2 - 2 x + 1)/(5 x^2 + 4).
- ZW-LIMITS-AT-INFINITY-002 Compute lim _x→∞ (2 x + 1)/(x^2 + 3).
- ZW-LIMITS-AT-INFINITY-003 Compute lim _x→∞ (4 x^3 - x)/(x^2 + 1).
- ZW-LIMITS-AT-INFINITY-004 Compute lim _x→∞ (5 x - 3)/(2 x + 7).
- ZW-LIMITS-AT-INFINITY-006 Compute lim _x→∞ (x^2 + x)/(3 x^2 - 1).
- ZW-LIMITS-AT-INFINITY-007 Compute lim _x→∞ (x^2 - 3 x)/(2 x + 1).
- ZW-LIMITS-AT-INFINITY-008 Compute lim _x→∞ (√4 x^2 + x - 2 x).
- ZW-LIMITS-AT-INFINITY-009 Compute lim _x→∞ (1)/(x).