91Ó°ÊÓ

Simplify the quotient \((f(x)-f(a)) /(x-a)\), and then guess the slope of the line tangent to the graph of \(f\) at \((a, f(a))\) \(f(x)=x^{2}+2 x ; a=-2\)

Suppose that \(x^{4} \leq f(x) \leq 2 x^{4}\) for all \(x\) in \([-1,1]\). Find \(\lim _{x \rightarrow 0} f(x) / x^{2}\).

Decide which of the given one-sided or two-sided limits exist as numbers, which as \(\infty\), which as \(-\infty\), and which do not exist. Where the limit is a number, evaluate it. $$ \lim _{x \rightarrow 0^{-}} \sqrt{e^{x}-1} $$

Show that the equation has at least one solution. $$ e^{-x}=x $$

Simplify the quotient \((f(x)-f(a)) /(x-a)\), and then guess the slope of the line tangent to the graph of \(f\) at \((a, f(a))\) \(f(x)=2 x^{2}+1 ; a=4\)

Simplify the quotient \((f(x)-f(a)) /(x-a)\), and then guess the slope of the line tangent to the graph of \(f\) at \((a, f(a))\) \(f(x)=x^{3}+1 ; a=-2\)

Decide which of the given one-sided or two-sided limits exist as numbers, which as \(\infty\), which as \(-\infty\), and which do not exist. Where the limit is a number, evaluate it. $$ \lim _{x \rightarrow 1^{-}} \ln (\ln x) $$

a. Draw the graph of a function \(f\) that satisfies the condition \(1+x

Solve the inequality for \(x\). $$ (2-x)^{2}(40-8 x)>0 $$

Decide which of the given one-sided or two-sided limits exist as numbers, which as \(\infty\), which as \(-\infty\), and which do not exist. Where the limit is a number, evaluate it. $$ \lim _{x \rightarrow 1^{+}} \ln (\ln x) $$