Limits and Continuity - Exercises

Below you will find the exercises we suggest you work on in connection with Limits and Continuity theme and the project problems which are to be done for the exercise classes 26.-30. August.

Section 1.1: Examples of Velocity, Growth Rate, and Area
1.1.5-8, 1.1.12-13.

Section 1.2: Limits of Functions
1.2.1, 1.2.3, 1.2.5, 1.2.9, 1.2.23, 1.2.25, 1.2.55, 1.2.57, 1.2.75 , 1.2.77.

Section 1.3: Limits at Infinity and Infinite Limits
1.3.3, 1.3.13, 1.3.29, 1.3.33.

Section 1.4: Continuity
1.4.1, 1.4.3, 1.4.5 ,1.4.11, 1.4.15, 1.4.17, 1.4.23, 1.4.29, 1.4.31.

Section 1.5: The Formal Definition of Limit
1.5.1, 1.5.5, 1.5.9, 1.5.20.

These problems are to be presented 26.-30. August. See here for where you should meet, and here to sign up for presenting a problem.

Problem 1

Define the function \(f\) by

\[f(x) = \begin{cases} 1 & \text{for} \,\, x \,\, \text{rational} \\ 0 & \text{for} \,\, x \,\, \text{irrational.} \end{cases} \]

At which points is \(f\) continuous/discontinuous?

Hint 1: You will have to use the \(\varepsilon\)-\(\delta\) definition of continuity.

Hint 2: Use the following property of the real numbers: For any two rational numbers \(x<y\), there exists an irrational number \(p\) such that \(x<p<y\). Similarly, for any two irrational numbers \(x<y\), there exists a rational number \(r\) such that \(x<r<y\). (Remark: From this property, one might think that there are an equal amount of rational and irrational number, but, in a precise sense, the set of irrational numbers is much, much larger than the set of rational numbers!)

Problem 2

Suppose \(f\) is a continuous function on the interval \([0,1]\), and \(f(0)=f(1)\).

a) Show that \(f(a)=f(a+1/2)\) for some \(a\in [0,1/2]\).

Hint: Let \(g(x)=f(x+1/2)-f(x)\), and use the Intermediate-Value Theorem.

b) If \(n\) is an integer larger than 2, show that \(f(a)=f(a+1/n)\) for some \(a\in [0,1-1/n]\),

Problem 3

Consider the functions \(f\) and \(g\) defined by \[f(x)=\frac{a\cos(x)+bx}{cx}\] for real numbers \(a,b,c\) so that \(c \neq 0\) and \[g(x)=\sqrt{(ax)^2+bx+c\sin(x)}-ax\] for real numbers \(a,b,c\) so that \(a>0\)

By using Maple, make plots of the functions \(f\) and \(g\) for different values of \(a,b,c\) and try to guess what the limits are when \(x\to\infty\). Also try to compute the limits in Maple.

Finally, calculate the limits by hand (without using L'Hosptal's rule). Hint: Use the Squeeze Theorem.

Problem 4

Let \(f\) be a continuous function, with domain the real numbers.

a) Given a closed interval \([a,b]\). Show that the set \(A = f([a,b]) = \{f(x) \, | \, x \in [a,b]\}\), i.e. the set of all numbers \(f(x)\) where \(x\) ranges over \([a,b]\), is also a closed interval.

b) Let \((a,b)\) be an open interval. Define the set of real numbers \(B = \{ x \, | \, f(x) \in (a,b) \}\), i.e. the set of all points \(x\) such that \(f(x)\) is a number in \((a,b)\). Suppose \(c\) is in \(B\). Show that there is an open interval \((c-h, c+h)\) centered in \(c\) which is a subset of \(B\), i.e. all numbers in \((c-h,c+h)\) are also in \(B\). (Sets like \(B\), such that if it includes some point, it also contains an open interval centered in that point, are called open sets.)

Hint: use the \(\varepsilon\)-\(\delta\) definition of continuity of \(f\) at the point \(c\).

c) Find a discontinuous function which doesn't satisfy the property in (a). Same for (b).