Problem 20
In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow 0}(1+x)^{1 / x} $$
Problem 20
(a) Let \(f_{1}\) and \(f_{2}\) be continuous at \(x_{0}\) and define $$ F(x)=\max \left(f_{1}(x), f_{2}(x)\right) $$ Show that \(F\) is continuous at \(x_{0}\) (b) Let \(f_{1}, f_{2}, \ldots, f_{n}\) be continuous at \(x_{0}\) and define $$ F(x)=\max \left(f_{1}(x), f_{2}(x), \ldots, f_{n}(x)\right) $$ Show that \(F\) is continuous at \(x_{0}\).
Problem 21
In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow \infty} x^{\sin (1 / x)} $$
Problem 21
We say that \(f\) has at least \(n\) zeros, counting multiplicities, on an interval \(I\) if there are distinct points \(x_{1}, x_{2}, \ldots, x_{p}\) in \(I\) such that $$ f^{(j)}\left(x_{i}\right)=0, \quad 0 \leq j \leq n_{i}-1, \quad 1 \leq i \leq p $$ and \(n_{1}+\cdots+n_{p}=n\). Prove: If \(f\) is differentiable and has at least \(n\) zeros, counting multiplicities, on an interval \(I,\) then \(f^{\prime}\) has at least \(n-1\) zeros, counting multiplicities, on \(I\).
Problem 21
Define (a) \(\lim _{x \rightarrow x_{0}} f(x)=\infty\) (b) \(\lim _{x \rightarrow x_{0}} f(x)=-\infty\)
Problem 21
In Evercises \(2.5 .19-2.5 .22, \Delta\) is the forwand difference operator with
spacing \(h>0\).
Let \(f^{\prime \prime \prime}\) be bounded on an open interval containing
\(x_{0}\) and \(x_{0}+2 h .\) Find a constant \(k\) such that the magnitude of the
error in the approximation
$$
f^{\prime}\left(x_{0}\right) \approx \frac{\Delta f\left(x_{0}\right)}{h}+k
\frac{\Delta^{2} f\left(x_{0}\right)}{h^{2}}
$$
is not greater than \(M h^{2}\), where \(M=\sup \left\\{\left|f^{\prime \prime
\prime}(c)\right||| x_{0}
Problem 21
Find the domains of \(f \circ g\) and \(g \circ f\). (a) \(f(x)=\sqrt{x}, \quad g(x)=1-x^{2}\) (b) \(f(x)=\log x, \quad g(x)=\sin x\) (c) \(f(x)=\frac{1}{1-x^{2}}, \quad g(x)=\cos x\) (d) \(f(x)=\sqrt{x}, \quad g(x)=\sin 2 x\)
Problem 22
In Evercises \(2.5 .19-2.5 .22, \Delta\) is the forwand difference operator with
spacing \(h>0\).
Prove: If \(f^{(n+1)}\) is bounded on an open interval containing \(x_{0}\) and
\(x_{0}+n h\), then
$$
\left|\frac{\Delta^{n}
f\left(x_{0}\right)}{h^{n}}-f^{(n)}\left(x_{0}\right)\right| \leq A_{n}
M_{n+1} h
$$
where \(A_{n}\) is a constant independent of \(f\) and
$$
M_{n+1}=\sup _{x_{0}
Problem 22
Give an example of a function \(f\) such that \(f^{\prime}\) exists on an interval \((a, b)\) and has a jump discontinuity at a point \(x_{0}\) in \((a, b),\) or show that there is no such function.
Problem 22
Find \(\begin{array}{ll}\text { (a) } \lim _{x \rightarrow 0} \frac{1}{x^{3}} & \text { (b) } \lim _{x \rightarrow 0} \frac{1}{x^{6}} \\ \text { (c) } \lim _{x \rightarrow x_{0}} \frac{1}{\left(x-x_{0}\right)^{2 k}} & \text { (d) } \lim _{x \rightarrow x_{0}} \frac{1}{\left(x-x_{0}\right)^{2 k+1}}\end{array}\) \((k=\) positive integer)
