Limits and Continuity: math test, College – download the PDF

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7 October 20267 October 2026

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Math tests College : Limits and Continuity — Zyro the alien explorer of Planète Maths

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

Sommaire

1 Computing limits / 4 pts

Compute: (a) \(\displaystyle\lim_{x\to5}\dfrac{x^2-25}{x-5}\); (b) \(\displaystyle\lim_{x\to0}\dfrac{\sqrt{x+4}-2}{x}\); (c) \(\displaystyle\lim_{x\to\infty}\dfrac{7x^2-x}{2x^2+9}\); (d) \(\displaystyle\lim_{x\to1}\dfrac{x^2+x-2}{x-1}\).

2 A jump / 3 pts

Let \(g(x)=2x-1\) for \(x<3\) and \(g(x)=10-x\) for \(x\ge3\). (a) Find \(\displaystyle\lim_{x\to3^-}g(x)\) and \(\displaystyle\lim_{x\to3^+}g(x)\). (b) Does \(\displaystyle\lim_{x\to3}g(x)\) exist? (c) Find \(g(3)\) and classify the discontinuity at 3.

3 Asymptotes and a hole / 3 pts

Let \(r(x)=\dfrac{2x+6}{x^2-9}\). (a) Find the vertical asymptote. (b) Find the location of the hole. (c) Find the horizontal asymptote.

4 Squeeze / 3 pts

For all \(x>0\), \(\dfrac{3}{x^2}\le\dfrac{4+\sin x}{x^2}\le\dfrac{5}{x^2}\). Use this to find \(\displaystyle\lim_{x\to+\infty}\dfrac{4+\sin x}{x^2}\). Justify.

5 Continuity at a point / 4 pts

Let \(f(x)=\dfrac{x^2-x-6}{x-3}\) for \(x\neq3\) and \(f(3)=k\). (a) Find \(\displaystyle\lim_{x\to3}f(x)\). (b) If \(k=1\), what type of discontinuity does \(f\) have at 3? (c) Find \(k\) so that \(f\) is continuous at 3.