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

1 Guess a limit from a table ★★★

Let \(f(x)=2x+1\). Compute \(f(2.9)\), \(f(2.99)\), \(f(2.999)\), \(f(3.1)\), \(f(3.01)\), and \(f(3.001)\). What limit do you suggest for \(f(x)\) as \(x\to3\)?

2 Direct substitution ★★★

Compute \(\displaystyle\lim_{x\to-2}\,(x^2+3x-1)\).

3 Applying the limit laws ★★★

As \(x\to a\), \(f(x)\to4\) and \(g(x)\to-2\). Find the limit of: (a) \(3f(x)+g(x)\); (b) \(f(x)\,g(x)\); (c) \(\dfrac{f(x)}{g(x)}\); (d) \((f(x))^2-g(x)\).

4 Reading limits on a graph ★★★

The graph of \(h\) is shown below. Open circles are missing points.

1. Find \(\displaystyle\lim_{x\to1^-}h(x)\) and \(\displaystyle\lim_{x\to1^+}h(x)\).
2. Does \(\displaystyle\lim_{x\to1}h(x)\) exist?
3. What is \(h(1)\)?

5 Two sides of the absolute value ★★★

Find (a) \(\displaystyle\lim_{x\to0^-}\dfrac{|x|}{x}\) and \(\displaystyle\lim_{x\to0^+}\dfrac{|x|}{x}\); (b) \(\displaystyle\lim_{x\to2^-}\dfrac{x-2}{|x-2|}\).

6 A simple limit at infinity ★★★

Compute \(\displaystyle\lim_{x\to+\infty}\dfrac{4x+7}{2x-1}\).

7 True or false? ★★★

True or false: “If \(f(2)\) is undefined, then \(\lim_{x\to2}f(x)\) does not exist.” Justify with an example.

8 Cancel a common factor ★★★

Compute \(\displaystyle\lim_{x\to-3}\dfrac{x^2+5x+6}{x+3}\).

9 Rationalize the numerator ★★★

Compute \(\displaystyle\lim_{x\to0}\dfrac{\sqrt{x+9}-3}{x}\).

10 Find all asymptotes ★★★

Let \(r(x)=\dfrac{6x^2+1}{3x^2-x}\). Find its horizontal and vertical asymptotes.

11 Square roots at infinity ★★★

Compute \(\displaystyle\lim_{x\to+\infty}\dfrac{\sqrt{4x^2+1}}{x+3}\) and \(\displaystyle\lim_{x\to-\infty}\dfrac{\sqrt{4x^2+1}}{x+3}\).

12 Is it continuous at 2? ★★★

Let \(f(x)=x^2+1\) for \(x<2\) and \(f(x)=3x-1\) for \(x\ge2\). Is \(f\) continuous at \(x=2\)? Check the three conditions.

13 Repair a hole ★★★

Let \(f(x)=\dfrac{x^2-16}{x-4}\) for \(x\neq4\) and \(f(4)=k\). Find \(k\) so that \(f\) is continuous at 4.

14 Parking garage fees ★★★

A garage charges 4 dollars for each hour or part of an hour, so the cost for parking \(t\) hours (\(0

15 Squeeze with a cosine ★★★

Use the squeeze theorem to find \(\displaystyle\lim_{x\to0}x^2\cos\dfrac4x\).

16 Squeeze at infinity ★★★

Find (a) \(\displaystyle\lim_{x\to+\infty}\dfrac{\sin x}{x}\) and (b) \(\displaystyle\lim_{x\to+\infty}\dfrac{2x+\cos x}{x}\).

17 Locate three roots ★★★

Let \(f(x)=x^3-4x+1\). Use the IVT to show that \(f\) has a root in each of \((-3,-2)\), \((0,1)\), and \((1,2)\). Why are there no other roots?

18 Classify the discontinuities ★★★

Let \(f(x)=\dfrac{x^2-1}{x^2-3x+2}\). Find where \(f\) is undefined and classify each discontinuity.

19 Make a piecewise function continuous ★★★

Find the constants \(a\) and \(b\) so that \(f\) is continuous everywhere, where \(f(x)=x+1\) for \(x<0\), \(f(x)=ax+b\) for \(0\le x\le3\), and \(f(x)=10-x\) for \(x>3\).

20 Cooling coffee ★★★

A cup of coffee in a room at 72 degrees F cools according to \(T(t)=72+98e^{-0.1t}\), where \(t\) is in minutes. (a) Find \(T(0)\) and \(T(10)\) to the nearest tenth. (b) Find \(\displaystyle\lim_{t\to+\infty}T(t)\) and the horizontal asymptote. Convert it to Celsius. (c) After how many minutes does the coffee reach 73 degrees F (nearest hundredth)?

21 When the IVT applies ★★★

(a) \(f\) is continuous on \([0,2]\) with \(f(0)=-3\) and \(f(2)=5\). Must \(f\) take the value 4? The value 6? (b) Let \(s(x)=-1\) for \(x<1\) and \(s(x)=1\) for \(x\ge1\). Show that \(s(0)<0