Evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{x \rightarrow 4} \dfrac{x^2 + 3 x}{x^2 - x - 12}\)

If \(2 x \leq g(x) \leq x^4 - x^2 + 2\) for all \(x\), evaluate \(\operatorname*{lim}\limits_{x \rightarrow 1} g(x)\).

Evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{t \rightarrow 0} \left(\dfrac{1}{t} - \dfrac{1}{t^2 + t}\right)\)

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} \dfrac{x^2 - x}{x^2 - 1} & \text{if } x \neq 1 \\ 1 & \text{if } x = 1 \end{cases}\), \(a = 1\)

Use continuity to evaluate the limit. \(\operatorname*{lim}\limits_{x \rightarrow 2} x \sqrt{20 - x^2}\)

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(F(x) = \sin(\cos(\sin x))\)

Which of the following functions \(f\) has a removable discontinuity at \(a\)? If the discontinuity is removable, find a function \(g\) that agrees with \(f\) for \(x \neq a\) and is continuous at \(a\).

The toll \(T\) charged for driving on a certain stretch of a toll road is \$5 except during rush hours (between 7 AM and 10 AM and between 4 PM and 7 PM) when the toll is \$7.