91Ó°ÊÓ

In Exercises \(74-80\), evaluate the one-sided limits. $$ \lim _{x \rightarrow 3^{-}}\left(\frac{6+x^{2}}{5}+\sqrt{13-2 x}\right) $$

In Exercises \(74-80\), evaluate the one-sided limits. $$ \lim _{x \rightarrow 4^{-}} \frac{x+\sqrt{16-3 x}}{x+\sqrt{16-x^{2}}} $$

In Exercises \(77-80\), evaluate the given limit. $$ \lim _{n \rightarrow \infty}\left(\frac{n+1}{n}\right)^{n} $$

In Exercises \(74-80\), evaluate the one-sided limits. $$ \lim _{x \rightarrow 0^{+}} \frac{\sqrt{\sin (x)}}{\sqrt{x}} $$

Let \(f:[0,1] \rightarrow[0,1]\) be a continuous function. Prove that there is a number \(c, 0 \leq c \leq 1,\) such that \(f(c)=c .\) Such a value is said to be a fixed point of \(f\). (Hint: Think about the function \(g(x)=f(x)-x .)\)

Here is an interesting way to use a sequence to solve a quadratic equation: Consider the equation \(x^{2}-\) \(6 x+5=0 .\) Write \(x^{2}=6 x-5,\) or \(x=6-5 / x .\) For the \(x\) on the right, substitute \(6-5 / x .\) The result is $$ x=6-\frac{5}{6-5 / x} $$ Again, for the \(x\) on the right, substitute \(x=6-5 / x\). The result is $$ x=6-\frac{5}{6-\frac{5}{6-5 / x}} $$ This process, which we may continue indefinitely, suggests that \(x\) is the limit of the sequence $$ 6,6-\frac{5}{6}, 6-\frac{5}{6-5 / 6}, 6-\frac{5}{6-\frac{5}{6-5 / 6}}, \ldots $$ This is called a continued fraction expansion. Compute the first 20 terms of this sequence. Do they seem to be converging to something? Is the limit a root of the original quadratic equation? Can you discover the other root in the same way?

Evaluate the given limit. $$ \lim _{n \rightarrow \infty}\left(\frac{n}{n+1}\right)^{n} $$

In Exercises \(74-80\), evaluate the one-sided limits. $$ \lim _{x \rightarrow 0^{+}} \frac{\sqrt{\sin (x)}}{\sqrt{x}} $$

Evaluate the given limit. $$ \lim _{n \rightarrow \infty}\left(1+\frac{3}{n}\right)^{2 n} $$

In Exercises \(74-80\), evaluate the one-sided limits. $$ \lim _{x \rightarrow 1^{-}} \frac{(\sqrt{x}-1) \sqrt{x^{2}-3 x+2}}{(1-x)^{3 / 2}} $$