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Magazine Problem 220
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. $$\lim _{x \rightarrow 1} \frac{x^{2}-1}{\sqrt{x}-1}$$
Problem 221
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. $$\lim _{x \rightarrow 4} \frac{4-x}{\sqrt{x}-2}$$
Problem 222
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. $$\lim _{x \rightarrow 4} \frac{1}{\sqrt{x}-2}$$
Problem 223
In the following exercises, use the squeeze theorem to prove the limit. $$\lim _{x \rightarrow 0} x^{2} \cos (2 \pi x)=0$$
Problem 224
In the following exercises, use the squeeze theorem to prove the limit. $$\lim _{x \rightarrow 0} x^{3} \sin \left(\frac{\pi}{x}\right)=0$$
Problem 225
Determine the domain such that the function \(f(x)=\sqrt{x-2}+x e^{x}\) is continuous over its domain.
Problem 225
In the following exercises, use the squeeze theorem to prove the limit. Determine the domain such that the function \(f(x)=\sqrt{x-2}+x e^{x}\) is continuous over its domain.
Problem 226
In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous. $$f(x)=\left\\{\begin{array}{l}{x^{2}+1, x>c} \\ {2 x, x \leq c}\end{array}\right.$$
Problem 227
In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous. $$f(x)=\left\\{\begin{array}{l}{\sqrt{x+1}, x>-1} \\ {x^{2}+c, x \leq-1}\end{array}\right.$$
Problem 228
In the following exercises, use the precise definition of limit to prove the limit. $$\lim _{x \rightarrow 1}(8 x+16)=24$$
