91Ó°ÊÓ

In the following exercises, write the appropriate \(\varepsilon-\delta\) definition for each of the given statements. $$\lim _{x \rightarrow a} f(x)=N$$

In the following exercises, write the appropriate \(\varepsilon-\delta\) definition for each of the given statements. $$\lim _{t \rightarrow b} g(t)=M$$

In the following exercises, write the appropriate \(\varepsilon-\delta\) definition for each of the given statements. $$\lim _{x \rightarrow c} h(x)=L$$

In the following exercises, write the appropriate \(\varepsilon-\delta\) definition for each of the given statements. $$\lim _{x \rightarrow a} \varphi(x)=A$$

The following graph of the function \(f\) satisfies $$\lim _{x \rightarrow 3} f(x)=-1 .$$ In the following exercises, determine a value of \(\delta>0\) that satisfies each statement. If \(0<|x-3|<\delta,\) then \(|f(x)+1|<1\)

[T] In the following exercises, use a graphing calculator to find a number \(\delta\) such that the statements hold true. \(\left|\sin (2 x)-\frac{1}{2}\right|<0.1,\) whenever \(\left|x-\frac{\pi}{12}\right|<\delta\).

\([\mathrm{T}]\) In the following exercises, use a graphing calculator to find a number \(\delta\) such that the statements hold true. \(\left|\sin (2 x)-\frac{1}{2}\right|<0.1,\) whenever \(\left|x-\frac{\pi}{12}\right|<\delta\)

[T] In the following exercises, use a graphing calculator to find a number \(\delta\) such that the statements hold true. \(|\sqrt{x-4}-2|<0.1,\) whenever \(|x-8|<\delta\).

\([\mathrm{T}]\) In the following exercises, use a graphing calculator to find a number \(\delta\) such that the statements hold true. \(|\sqrt{x-4}-2|<0.1,\) whenever \(|x-8|<\delta\)

In the following exercises, use the precise definition of limit to prove the given limits. $$\lim _{x \rightarrow 2}(5 x+8)=18$$