Chapter 1: Problem 12
Compare the given number with the number \(e\). Is the number less than or greater than \(e\) ? $$ 1+1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040} $$
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Boyle's Law For a quantity of gas at a constant temperature, the pressure \(P\) is inversely proportional to the volume \(V\). Find the limit of \(P\) as \(V \rightarrow 0^{+}\).
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=x^{n}\) where \(n\) is odd, then \(f^{-1}\) exists.
Show that the function \(f(x)=\left\\{\begin{array}{ll}0, & \text { if } x \text { is rational } \\ k x, & \text { if } x \text { is irrational }\end{array}\right.\) is continuous only at \(x=0\). (Assume that \(k\) is any nonzero real number.)
Prove that if \(\lim _{x \rightarrow c} f(x)=0\) and \(|g(x)| \leq M\) for a fixed number \(M\) and all \(x \neq c,\) then \(\lim _{x \rightarrow c} f(x) g(x)=0\).
Prove that a function has an inverse function if and only if it is one-to-one
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