Problem 15
In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow 0}(\cot x-\csc x) $$
Problem 15
(a) Prove: If \(f\) is continuous at \(x_{0}\) and \(f\left(x_{0}\right)>\mu,\) then \(f(x)>\mu\) for all \(x\) in some neighborhood of \(x_{0}\). (b) State a result analogous to (a) for the case where \(f\left(x_{0}\right)<\mu\). (c) Prove: If \(f(x) \leq \mu\) for all \(x\) in \(S\) and \(x_{0}\) is a limit point of \(S\) at which \(f\) is continuous, then \(f\left(x_{0}\right) \leq \mu\). (d) State results analogous to (a), (b), and (c) for the case where \(f\) is continuous from the right or left at \(x_{0}\).
Problem 16
Find an upper bound for the magnitude of the error in the approximation.
(a) \(\sin x \approx x, \quad|x|<\frac{\pi}{20}\)
(b) \(\sqrt{1+x} \approx 1+\frac{x}{2}, \quad|x|<\frac{1}{8}\)
(c) \(\cos x \approx
\frac{1}{\sqrt{2}\left[1-\left(x-\frac{\pi}{4}\right)\right] .
\frac{\pi}{4}
Problem 16
Show that \(f(a+)\) and \(f(b-)\) exist (finite) if \(f^{\prime}\) is bounded on \((a, b)\). HINT: See Exercise \(2.1 .38 .\)
Problem 16
Let \(|f|\) be the function whose value at each \(x\) in \(D_{f}\) is \(|f(x)| .\) Prove: If \(f\) is continuous at \(x_{0},\) then so is \(|f|\). Is the converse true?
Problem 16
In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow 0}\left(\frac{1}{\sin x}-\frac{1}{x}\right) $$
Problem 17
Suppose that \(f\) is continuous on \([a, b], f_{+}^{\prime}(a)\) exists, and \(\mu\) is between \(f_{+}^{\prime}(a)\) and \((f(b)-f(a)) /(b-a)\). Show that \(f(c)-f(a)=\mu(c-a)\) for some \(c\) in \((a, b)\).
Problem 17
Prove: If
$$
T_{n}(x)=\sum_{r=0}^{n} \frac{x^{r}}{r !}
$$
then
$$
T_{n}(x)
Problem 17
In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow \pi}|\sin x|^{\tan x} $$
Problem 17
Prove: If \(f\) is monotonic on \([a, b]\), then \(f\) is piecewise continuous on \([a, b]\) if and only if \(f\) has only finitely many discontinuities in \([a, b]\).
