5.4: Exercises
EXERCISE 5.1. Prove that the definitions of limit on pages 129 and 130 are the same.
EXERCISE 5.2. Prove Lemma 5.14, and the related assertion that \(|c|-|d| \leq|c+d|\) .
EXERCISE 5.3. For \(n \in \mathbb{N}^{+}, a_{i} \in \mathbb{R}(1 \leq i \leq n)\) , prove that \[\left|\sum_{i=1}^{n} a_{i}\right| \leq \sum_{i=1}^{n}\left|a_{i}\right| .\] ExERCISE 5.4. Prove Lemma \(5.15\) .
EXERCISE 5.5. Prove part (v) of Theorem 5.9.
EXERCISE 5.6. Give an example of two functions \(f\) and \(g\) that don’t have limits at a point \(a\) but such that \(f+g\) does. For the same pair of functions, can \(f-g\) also have a limit at \(a\) ? EXERCISE 5.7. Assume that \(f\) is a real function and \(\lim _{x \rightarrow a} f(x)=\) \(L\) . Prove that if \(X \subseteq \operatorname{Dom}(f)\) , then \[\lim _{X \ni x \rightarrow a} f(x)=L .\] EXERCISE 5.8. Use Archimedes’s method (the method of Riemann sums) to prove that \[\int_{0}^{1} x^{2} d x=\frac{1}{3}\] (You will need to know a formula for \(\sum_{k=0}^{n} k^{2}\) - see Proposition 4.6).
EXERCISE 5.9. Use Archimedes’s method to prove that \[\int_{0}^{1} x^{3} d x=\frac{1}{4} .\] (See Exercise 4.16).
EXERCISE 5.10. Prove that the Heaviside function has both left and right-hand limits at \(0 .\)
EXERCISE 5.11. Prove that a function has a limit at a point if and only if it has both left and right limits at that point and their values coincide.
EXERCISE 5.12. Prove that Theorem \(5.9\) applies to restricted limits.
EXERCISE 5.13. The point \(a\) is a limit point of the set \(X\) if, for every \(\delta>0\) , there exists a point \(x\) in \(X \backslash\{a\}\) with \(|x-a|<\delta\) . Let \(f\) be a real-valued function on \(X \subseteq \mathbb{R}\) . Prove that if \(a\) is a limit point of \(X\) , then if \(f\) has a restricted limit at \(a\) it is unique. Prove that if \(a\) is not a limit point of \(X\) , then every real number is a restricted limit of \(f\) at \(a\) .
EXERCISE 5.14. Prove that \(\lim _{x \rightarrow 0} \sin (x) / x=1\) .
EXERCISE 5.15. Prove Proposition 5.22.
EXERCISE 5.16. Prove that the function \(f(x)=x\) is continuous everywhere on \(\mathbb{R}\) . ExERCISE 5.17. A formula for the Fibonacci numbers is given in Exercise 4.12. Evaluate \(\lim _{n \rightarrow \infty} F_{n+1} / F_{n}\) .
EXERCISE 5.18. How large must \(n\) be to ensure that \(F_{n+1} / F_{n}\) is within \(10^{-1}\) of the limit in Exercise 5.17? Within \(10^{-2}\) ? Within \(10^{-k}\) ?
EXERCISE 5.19. Define the function \(\psi: \mathbb{R} \rightarrow \mathbb{R}\) by \[\psi(x):= \begin{cases}0 & x \notin \mathbb{Q} \\ 1 & x \in \mathbb{Q} .\end{cases}\] Prove that \(\psi\) is discontinuous everywhere.
EXERCISE \(5.20\) . Define the function \(\phi: \mathbb{R} \rightarrow \mathbb{R}\) by \[\phi(x):= \begin{cases}0 & x \notin \mathbb{Q} \\ \frac{1}{n} & x \in \mathbb{Q} \backslash\{0\}, x=\frac{m}{n}, \operatorname{gcd}(m, n)=1, n>0 \\ 1 & x=0 .\end{cases}\] Prove that \(\phi\) is continuous at every irrational number and discontinuous at every rational number.
EXERCISE 5.21. Prove that a real-valued function \(f\) on an open interval \(I\) is continuous at any point where its derivative exists, i.e. where \[\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\] exists. What is the converse of this assertion? Prove that the converse is not true.
EXERCISE 5.22. Prove that if the function \(f\) has the limit \(L\) from the right at \(a\) , then the sequence \(f\left(a+\frac{1}{n}\right)\) has limit \(L\) as \(n \rightarrow \infty\) . Show that the converse is false in general.
EXERCISE 5.23. Let \(f\) and \(g\) be real functions. Let \(a \in \mathbb{R}\) and suppose that \[\lim _{x \rightarrow a} g(x)=L_{1}\] and \[\lim _{x \rightarrow L_{1}} f(x)=L_{2} .\] Prove that \[\lim _{x \rightarrow a} f \circ g=L_{2} .\] If \(g\) is continuous at \(a\) and \(f\) is continuous at \(g(a)\) , is \(f \circ g\) continuous at \(a\) ?
EXERCISE 5.24. Let \(f\) be a real function, \(a \in \mathbb{R}\) and \(\lim _{x \rightarrow a} f(x)=\) \(L\) . If \(\left\langle a_{n}\right\rangle\) converges to \(a\) , prove that \(\left\langle f\left(a_{n}\right)\right\rangle\) converges to \(L\) .
EXERCISE 5.25. Complete Example 5.26. That is, prove that \[\lim _{n \rightarrow \infty}\left(\frac{1}{n^{3}}\right) \frac{(n)(n+1)(2 n+1)}{6}=1 / 3 .\] EXERCISE 5.26. Evaluate \[\lim _{n \rightarrow \infty} \sum_{k=0}^{n}\left(\frac{k}{n}\right) \frac{1}{n} .\] Can you give a geometrical interpretation of this limit?
EXERCISE 5.27. Use induction to prove that every polynomial is continuous at every real number.
EXERCISE 5.28. Let \(-1<x<1\) . Prove that the geometric series with ratio \(x, \sum_{k=0}^{\infty} x^{k}\) , converges to \(\frac{1}{1-x}\) .
EXERCISE 5.29. Let the Fibonacci numbers \(F_{n}\) be defined as in Exercise 4.12. Consider the power series \(F(x)=\sum_{n=1}^{\infty} F_{n} x^{n}\) . Prove that the power series satisfies \[F(x)=x^{2} F(x)+x F(x)+x .\] Solve (5.32) for \(F(x)\) , decompose it by partial fractions, and use Exercise \(5.28\) to derive Formula \(4.20\) . This technique to find a formula for \(F_{n}\) by studying the function \(F\) is often fruitful. The function \(F\) is called the generating function for the sequence.
EXERCISE 5.30. Suppose one defines a sequence with the same recurrence relation as the Fibonacci numbers, \(F_{n+2}=F_{n+1}+F_{n}\) , but with different starting values for \(F_{1}\) and \(F_{2}\) . Find the generating function for the new sequence, and hence calculate a formula for the general term. Is \(\lim _{n \rightarrow \infty} F_{n+1} / F_{n}\) always the same?
EXERCISE 5.31. Prove that sine and cosine are continuous functions on all of \(\mathbb{R}\) .