Functions, limits and continuity – university Year 1 maths worksheet with answers to download as PDF

Evaluate each limit, or explain why it does not exist. Say what technique you use.

1. \(\displaystyle\lim_{x\to2}\dfrac{x^2-4}{x-2}\) [2 marks]
2. \(\displaystyle\lim_{x\to0}\dfrac{\sqrt{1+x}-1}x\) [3 marks]
3. \(\displaystyle\lim_{x\to\infty}\dfrac{3x^2+x}{2x^2-5}\) [3 marks]
4. \(\displaystyle\lim_{x\to0}\dfrac{\sin3x}x\) [2 marks]

Use the \(\varepsilon\)-\(\delta\) definition of a limit to prove each statement. Do your rough work first and then write the proof forwards.

1. \(\displaystyle\lim_{x\to2}(3x-1)=5\) [3 marks]
2. \(\displaystyle\lim_{x\to1}x^2=1\) [5 marks]

Find the left-hand and right-hand limits at the stated point, and say whether the two-sided limit exists.

1. \(f(x)=\dfrac{|x|}x\) at \(x=0\) [3 marks]
2. \(g(x)=\begin{cases}x^2+1&x\le2\\ 7-x&x>2\end{cases}\) at \(x=2\) [3 marks]
3. \(k(x)=\dfrac1{x-1}\) at \(x=1\) [3 marks]

Find the constants that make each function continuous at the given point.

1. \(f(x)=\begin{cases}ax+3&x\le2\\ x^2+1&x>2\end{cases}\) at \(x=2\) [3 marks]
2. \(g(x)=\dfrac{x^2-1}{x-1}\) for \(x\ne1\) and \(g(1)=a\) [3 marks]

Use the intermediate value theorem to prove that each equation has a solution in the stated interval. Check that all the conditions hold.

1. \(x^3+x-1=0\) in \((0,1)\) [3 marks]
2. \(\cos x=x\) in \([0,1]\) [4 marks]

Answer each part.

1. Find the natural domain and the image of \(f(x)=\ln(x^2-1)\). [3 marks]
2. Show that \(g(x)=x^3+x\) is injective on \(\mathbb R\). [3 marks]
3. Is \(h(x)=x^2\) injective on \(\mathbb R\)? On \([0,\infty)\)? [2 marks]

An energy company charges for the first 200 kWh each month at 30p per kWh. Above 200 kWh, there is a fixed charge of £\(k\) plus 20p for every kWh used. The monthly bill in pounds is \(C(u)=\begin{cases}0.30u&0\le u\le200\\ k+0.20u&u>200\end{cases}\), where \(u\) is the number of kWh. The company wants the bill to have no sudden jump.

1. Find the bill for 200 kWh using the first rule.
2. Find \(k\) so that \(C\) is continuous at \(u=200\).
3. Find the bill for 350 kWh.
4. Explain what a jump in the tariff would mean for a customer using 200 kWh and for one using 201 kWh.

A post office charges for a letter of weight \(w\) grams according to the table.

1. Find the left-hand and right-hand limits of the price function \(P(w)\) at \(w=100\), and at \(w=250\).
2. Is \(P\) continuous at \(w=100\)? Classify the discontinuity.
3. What is \(P(100)\)? Is \(P\) left-continuous or right-continuous at \(w=100\)?
4. Sketch the graph in words and state the cost of sending a 250 g letter and a 251 g letter.

The temperature in a greenhouse, in \(^\circ\)C, \(t\) hours after midnight, is modelled by \(T(t)=9-6\cos\left(\dfrac{\pi t}{12}\right)\) for \(0\le t\le24\).

1. Show that \(T(6)=9\) and find \(T(14)\) to 1 decimal place.
2. Use the intermediate value theorem to show that the temperature is exactly \(12^\circ\)C at some time between 06:00 and 14:00.
3. Evaluate \(T(10)\) to locate the time more precisely.
4. Solve \(T(t)=12\) exactly and check against part (c).

A crate has a square base of side \(x\) metres and height \((x+1)\) metres. The volume must be 10 m\(^3\), so \(f(x)=x^3+x^2-10=0\).

1. Show that \(f(1)=-8\) and \(f(2)=2\) and explain why there is a solution in \((1,2)\).
2. Carry out three steps of the bisection method starting from \([1,2]\) and complete a table of midpoints and signs.
3. State the interval of length \(\frac18\) that contains the root.
4. After how many bisection steps starting from \([1,2]\) is the root guaranteed to be within 0.001?

A stone is dropped from a bridge. After \(t\) seconds it has fallen \(s=4.9t^2\) metres. The average speed between \(t=2\) and \(t=2+h\) is \(\dfrac{s(2+h)-s(2)}h\).

1. Show that the average speed is \(19.6+4.9h\).
2. Complete the table.
3. Find the limit as \(h\to0\) and explain what it means.
4. Find the speed after 3 seconds in the same way.

A machine cuts rods whose length in cm is \(L=3d-1\), where \(d\) is a dial setting. The setting \(d=2\) should give 5 cm. Rods are accepted if they are within \(\varepsilon\) cm of 5 cm.

1. Show that \(|L-5|=3|d-2|\).
2. How close to 2 must the dial be if \(\varepsilon=0.03\)?
3. How close for \(\varepsilon=0.003\)?
4. Explain how this is an example of the definition \(\displaystyle\lim_{d\to2}(3d-1)=5\).

The number of birds in a colony after \(t\) years on a protected island is modelled by \(P(t)=\dfrac{1200t}{t+3}\).

1. Find \(P(3)\) and \(P(27)\).
2. Find \(\displaystyle\lim_{t\to\infty}P(t)\) and interpret it.
3. Show that \(1200-P(t)=\dfrac{3600}{t+3}\).
4. Use part (c) to find how large \(t\) must be for the population to be within 1 bird of the limit.

The displacement in cm of a spring in a lab is \(y(t)=e^{-0.5t}\cos3t\) for \(t\ge0\) (in seconds).

1. Show that \(-e^{-0.5t}\le y(t)\le e^{-0.5t}\).
2. Use the squeeze theorem to find \(\displaystyle\lim_{t\to\infty}y(t)\).
3. After what time is \(|y(t)|\) certain to be below 0.01 cm?
4. Does \(y\) have a limit if the exponential factor is removed, so that \(y=\cos3t\)? Explain.

A railway engineer joins two pieces of track with a straight middle section. The height profile (in metres) as a function of \(x\) (in hundreds of metres) is \(f(x)=\begin{cases}2x&x<1\\ ax+b&1\le x\le3\\ x^2-1&x>3\end{cases}\).

1. Write the two equations that \(a\) and \(b\) must satisfy for \(f\) to be continuous at \(x=1\) and at \(x=3\).
2. Solve for \(a\) and \(b\).
3. Work out the gradients of the three pieces. Is the track smooth (differentiable) at \(x=1\) and \(x=3\)?
4. What would a train passenger feel at a point where the gradient changes suddenly? Suggest a better design.

A data logger records four quantities, each modelled by a function with a break at the stated point. Classify each discontinuity as removable, jump, infinite, or essential (oscillating).

1. Classify each discontinuity and justify your answer with limits.
2. For the removable one, say how to redefine the function so that it is continuous.
3. Which of the four cannot be repaired by changing the value at the single point?
4. Sketch (in words) the behaviour of \(f_4\) near 0, using the points \(x=\dfrac1{n\pi}\).

On Saturday, Zara walks up a mountain path from the car park (height 0 m) at 08:00 to the summit (height 1 200 m) at 16:00, and her height at time \(t\) is \(u(t)=18.75(t-8)^2\). On Sunday she walks down the same path, starting at the summit at 08:00 and reaching the car park at 16:00 with height \(d(t)=1200-150(t-8)\).

1. Check \(u(8)\), \(u(16)\), \(d(8)\) and \(d(16)\).
2. Let \(g(t)=u(t)-d(t)\). Show that \(g(8)<0\lt g(16)\) and explain why Zara is at the same height at the same time of day on both days at some moment.
3. Find that time by solving \(g(t)=0\).
4. Would the conclusion still hold if she rested on the way and her height were only piecewise continuous? Explain.

A computer scientist compares the number of steps of three algorithms with input size \(n\): \(A(n)=n^2+5n\), \(B(n)=3n^2-1\) and \(C(n)=2^n\).

1. Find \(\displaystyle\lim_{n\to\infty}\dfrac{A(n)}{B(n)}\) and interpret it.
2. Find \(\displaystyle\lim_{n\to\infty}\dfrac{n^2}{2^n}\) using l'Hôpital's rule twice (treat \(n\) as a real variable).
3. What does part (b) say about algorithm \(C\) compared with \(A\)?
4. Evaluate \(\displaystyle\lim_{n\to\infty}\left(\sqrt{n^2+n}-n\right)\) by multiplying by the conjugate.

A square metal plate has side \(x\) cm and area \(x^2\) cm\(^2\). The ideal side is 2 cm, so the ideal area is 4 cm\(^2\). The area must be within \(\varepsilon\) cm\(^2\) of 4.

1. Show that \(|x^2-4|=|x-2||x+2|\).
2. If \(|x-2|<1\) show that \(|x+2|<5\).
3. Prove, using \(\delta=\min\left(1,\dfrac\varepsilon5\right)\), that \(\displaystyle\lim_{x\to2}x^2=4\).
4. Find a suitable \(\delta\) when the area tolerance is \(\varepsilon=0.1\) cm\(^2\), and say what it means for the workshop.

Let \(T(t)\) be the temperature at time \(t\) hours (a continuous function) with \(T(0)=T(24)\), so the temperature at midnight is the same on two consecutive days. Prove that there are two times exactly 12 hours apart with the same temperature.

1. Define \(g(t)=T(t+12)-T(t)\) for \(0\le t\le12\). Why is \(g\) continuous?
2. Show that \(g(0)=-g(12)\).
3. Deduce that \(g(c)=0\) for some \(c\in[0,12]\).
4. Explain what this proves about the temperature, and why it works for any continuous function on a circle (such as the temperature around the equator at a given moment).