AP Calculus AB Introducing Limits Worksheet Solutions (PDF)
This limits intro answers for AP Calculus AB covers introducing limits (Common Core AP CALC AB 1.1, AP CALC AB 1.2, AP CALC AB 1.3, AP CALC AB 1.4), with a free PDF to print.
This answer key gives a full worked solution for every problem in the limits worksheet. Compare each step, not only the final answer, because AP graders award credit for a correct setup and clear justification. Final answers are in bold. Redo any problem you missed before moving on to the test.
AP Calculus AB · Common Core State Standards: AP CALC AB 1.1, AP CALC AB 1.2, AP CALC AB 1.3, AP CALC AB 1.4
Limits intro answers: exercises 1 to 10
Exercise 1 – Reading limit notation
The notation says that as \(x\) gets closer and closer to \(5\) from both sides, the outputs \(f(x)\) get closer and closer to \(12\).
Final answer: the limit of f(x) as x approaches 5 is 12.
Exercise 2 – Table for a linear function
Add 4 to each input: \(6.9, 6.99, 6.999, 7.001, 7.01, 7.1\).
The outputs approach \(7\) from both sides. Final answer: \(\lim_{x\to 3}(x+4)=7\).
Exercise 3 – Table with an undefined point
Evaluate each input: \(5.9, 5.99, 5.999, 6.001, 6.01, 6.1\).
Although \(g(3)\) is undefined, the outputs approach \(6\). Final answer: \(6\).
Exercise 4 – Limit versus function value
Follow the line from the left and from the right toward \(x=2\). Both sides approach height \(3\).
The filled point is at \((2,1)\), so the actual value is \(f(2)=1\).
Final answer: \(\lim_{x\to 2} f(x)=3\) and \(f(2)=1\).
Exercise 5 – One-sided limits from a graph
From the left, the graph follows \(y=2x-1\) and reaches height \(2(1)-1=1\). So the left-hand limit is \(1\).
From the right, it follows \(y=x+2\) and starts at \(1+2=3\). So the right-hand limit is \(3\).
The one-sided limits differ, so \(\lim_{x\to 1} f(x)\) does not exist.
Exercise 6 – Limit at a smooth point
The curve is unbroken at \(x=2\), so the outputs approach the point on the graph.
Compute \(f(2)=\frac{4}{2}-2=0\). Final answer: \(0\).
Exercise 7 – Absolute value quotient
For negative inputs, \(|x|=-x\), so the quotient is \(-1\). For positive inputs it is \(1\). Table outputs: \(-1, -1, -1, 1, 1, 1\).
The left-hand limit is \(-1\) and the right-hand limit is \(1\). Final answer: the limit does not exist because the one-sided limits differ.
Exercise 8 – Unbounded outputs
At \(x=\pm 0.5\) the output is \(4\); at \(x=\pm 0.1\) it is \(100\); at \(x=\pm 0.01\) it is \(10{,}000\). The outputs grow without bound from both sides.
Final answer: the outputs increase without bound, so the limit does not exist as a real number. In symbols, the limit is \(\infty\) (a vertical asymptote at \(x=0\)).
Exercise 9 – Sine over x
The outputs are \(0.99833, 0.99998, 0.99998, 0.99833\).
They approach \(1\) from both sides. Final answer: \(1\).
Exercise 10 – Square root difference
The outputs are \(0.25158, 0.25016, 0.25002, 0.24998, 0.24984, 0.24846\).
They approach \(0.25\) from both sides, so the limit is \(\frac{1}{4}\).
Exercises 11 and up
Exercise 11 – Average rate of change
\(\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4\).
\(\frac{f(1.1)-f(1)}{0.1}=\frac{1.21-1}{0.1}=2.1\). And \(\frac{1.0201-1}{0.01}=2.01\).
The averages approach \(2\). Final answer: \(4,\ 2.1,\ 2.01\); instantaneous rate about \(2\).
Exercise 12 – Limit from a table
As \(x\) approaches \(5\) from the left, \(g(x)\) is \(2.9, 2.99, 2.999\). From the right it is \(3.1, 3.01, 3.001\).
Both sides approach \(3\). The table lists \(g(5)=1\), a different number.
Final answer: \(\lim_{x\to 5} g(x)=3\) and \(g(5)=1\).
Exercise 13 – Piecewise one-sided limits
Left: use \(x^{2}\), so \(\lim_{x\to 1^-} f(x)=1^{2}=1\).
Right: use \(2x+1\), so \(\lim_{x\to 1^+} f(x)=2(1)+1=3\).
Final answer: left \(1\), right \(3\), and the two-sided limit does not exist.
Exercise 14 – Making a limit exist
Left-hand limit: \(3(2)+1=7\). Right-hand limit: \(2^{2}+k=4+k\).
Set them equal: \(4+k=7\), so \(k=3\).
Final answer: \(k=3\).
Exercise 15 – Square root near the edge
\(\sqrt{0.1}\approx 0.3162\), \(\sqrt{0.01}=0.1\), \(\sqrt{0.001}\approx 0.0316\). The outputs approach \(0\), so the right-hand limit is \(0\).
For \(x < 1\) the expression \(x-1\) is negative, so \(\sqrt{x-1}\) is not a real number. There are no inputs on the left to test.
Exercise 16 – Parking garage fee
Just below 2 hours (for example \(t=1.99\)), the fee is \$6. So \(\lim_{t\to 2^-} C(t)=6\).
Just above 2 hours (for example \(t=2.01\)), the fee is \$9. So \(\lim_{t\to 2^+} C(t)=9\).
Final answer: left \$6, right \$9, and the two-sided limit does not exist.
Exercise 17 – Speed of a skateboarder
\(\frac{s(3+h)-s(3)}{h}\) with \(s(3)=9\): \(\frac{16-9}{1}=7\), \(\frac{9.61-9}{0.1}=6.1\), \(\frac{9.0601-9}{0.01}=6.01\), \(\frac{9.006001-9}{0.001}=6.001\).
The averages approach \(6\). Final answer: about \(6\text{ m/s}\) at \(t=3\).
Exercise 18 – Cooling coffee
\(\frac{150-180}{2}=-15\), \(\frac{128-150}{2}=-11\), \(\frac{110-128}{2}=-9\), \(\frac{96-110}{2}=-7\) degrees per minute.
The coffee cools more slowly as time passes. Final answer: \(-15,\ -11,\ -9,\ -7\text{ °F/min}\); near \(t=8\) the rate is about \(-7\) °F per minute or a little slower.
Exercise 19 – Filling a water tank
\(V(5)=100\). Then \(\frac{V(5.1)-100}{0.1}=\frac{104.04-100}{0.1}=40.4\).
\(\frac{100.4004-100}{0.01}=40.04\) and \(\frac{100.040004-100}{0.001}=40.004\).
The averages approach \(40\). Final answer: \(40\text{ gallons per minute}\) at \(t=5\).
Exercise 20 – Hole in a rational graph
(a) At \(x=3\) the denominator is \(0\) and the numerator is \(9-3-6=0\), so \(g(3)=\frac{0}{0}\) is undefined.
(b) The outputs are \(4.9, 4.99, 5.01, 5.1\), approaching \(5\) from both sides.
(c) The graph follows the line \(y=x+2\) with a hole at \((3,5)\).
Final answer: \(\lim_{x\to 3} g(x)=5\) and a hole at \((3,5)\).
Exercise 21 – Several limits from one graph
- The parabola reaches height \(-0.5(1)^{2}+3=2.5\), so the left-hand limit is \(2.5\).
- The line starts at \(1-1.5=-0.5\) as \(x\to 1^+\), so the right-hand limit is \(-0.5\).
- The filled point is at \((1,-1)\), so \(f(1)=-1\).
- The one-sided limits differ, so the limit does not exist.
- The graph is unbroken at \(x=3\) with \(f(3)=3-1.5=1.5\), so the limit is \(1.5\).
Exercise 22 – Does the limit equal the value?
For \(x\neq 4\), \(h(x)=x+4\) (since \(x^{2}-16=(x-4)(x+4)\)). The table gives \(7.9, 7.99, 8.01, 8.1\), approaching \(8\).
But \(h(4)=2\). Since \(\lim_{x\to 4} h(x)=8\neq 2=h(4)\), the limit is not the function value.
Final answer: Jamal is wrong; the limit is \(8\) while \(h(4)=2\).
Key facts behind this limits intro answers
- A limit describes nearby behavior, not the value at the point itself.
- The two-sided limit exists only if the left-hand and right-hand limits are equal.
- Tables and graphs give estimates; algebra gives exact values.
- A limit fails to exist because of a jump, unbounded growth or oscillation.
How to use this limits intro answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
Download the limits intro answers as a PDF with the button below and keep it with your notes.
Download the limits intro answers as a PDF with the button below and keep it with your notes.
Download the limits intro answers as a PDF with the button below and keep it with your notes.
Download the limits intro answers as a PDF with the button below and keep it with your notes.
Download the limits intro answers as a PDF with the button below and keep it with your notes.
Download the limits intro answers as a PDF with the button below and keep it with your notes.
All AP Calculus AB chapters: AP Calculus AB math.
