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Magazine Problem 42
Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow 2}(-1) $$
Problem 42
Finding a Limit In Exercises \(41-46,\) write a simpler function that agrees with the given function at all but one point. Then find the limit of the function. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow 0} \frac{x^{4}-5 x^{2}}{x^{2}} $$
Problem 42
Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\cos \frac{\pi x}{2} $$
Problem 42
Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 3^{+}}\left(\frac{x}{3}+\cot \frac{\pi x}{2}\right) $$
Problem 43
Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 0^{+}} \frac{2}{\sin x} $$
Problem 43
Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow 0} \sqrt[3]{x} $$
Problem 43
Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\frac{x}{x^{2}-x} $$
Problem 43
Finding a Limit In Exercises \(41-46,\) write a simpler function that agrees with the given function at all but one point. Then find the limit of the function. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow-1} \frac{x^{2}-1}{x+1} $$
Problem 44
Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\frac{x}{x^{2}-4} $$
Problem 44
Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow(\pi / 2)^{+}} \frac{-2}{\cos x} $$
