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Magazine Problem 86
Find the limits in Exercises \(84-90 .\) (Hint: Try multiplying and dividing by the conjugate.) $$ \lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+3}+x\right) $$
Problem 87
In Exercises \(83-88,\) use a CAS to perform the following steps: a. Plot the function near the point \(c\) being approached. b. From your plot guess the value of the limit. $$\lim _{x \rightarrow 0} \frac{1-\cos x}{x \sin x}$$
Problem 87
Find the limits in Exercises \(84-90 .\) (Hint: Try multiplying and dividing by the conjugate.) $$ \lim _{x \rightarrow-\infty}\left(2 x+\sqrt{4 x^{2}+3 x-2}\right) $$
Problem 88
Find the limits in Exercises \(84-90 .\) (Hint: Try multiplying and dividing by the conjugate.) $$ \lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}-x}-3 x\right) $$
Problem 89
Find the limits in Exercises \(84-90 .\) (Hint: Try multiplying and dividing by the conjugate.) $$ \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+3 x}-\sqrt{x^{2}-2 x}\right) $$
Problem 90
Find the limits in Exercises \(84-90 .\) (Hint: Try multiplying and dividing by the conjugate.) $$ \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+x}-\sqrt{x^{2}-x}\right) $$
Problem 93
Use formal definitions to prove the limit statements in Exercises \(93-96\) $$ \lim _{x \rightarrow 0} \frac{-1}{x^{2}}=-\infty $$
Problem 94
Use formal definitions to prove the limit statements in Exercises \(93-96\) $$ \lim _{x \rightarrow 0} \frac{1}{|x|}=\infty $$
Problem 103
Graph the rational functions in Exercises \(103-108 .\) Include the graphs and equations of the asymptotes. $$ y=\frac{x^{2}}{x-1} $$
Problem 104
Graph the rational functions in Exercises \(103-108 .\) Include the graphs and equations of the asymptotes. $$ y=\frac{x^{2}+1}{x-1} $$
