Stewart 9th - 1.6, 1.8 (10Q)

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1 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{(h - 3)^2 - 9}{h} \)
2 Limit Laws - Algebraic · Level 2
\( \operatorname*{lim}\limits_{x \rightarrow 4} \dfrac{x^2 + 3 x}{x^2 - x - 12} \)
3 Limit Laws - Algebraic · Level 2
(a) What is wrong with the following equation? \(\dfrac{x^2 + x - 6}{x - 2} = x + 3\)
(b) In view of part (a), explain why the equation \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 + x - 6}{x - 2} = \operatorname*{lim}\limits_{x \rightarrow 2} (x + 3)\) is correct.

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4 Limit Laws - Squeeze Theorem · Level 2
If \(2 x \leq g(x) \leq x^4 - x^2 + 2\) for all \(x\), evaluate \(\operatorname*{lim}\limits_{x \rightarrow 1} g(x)\).
5 Limit Laws - Algebraic · Level 3
\( \operatorname*{lim}\limits_{t \rightarrow 0} \left(\dfrac{1}{t} - \dfrac{1}{t^2 + t}\right) \)
6 Continuity - Discontinuity Analysis · Level 3
Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} \dfrac{x^2 - x}{x^2 - 1} & \quad \text{if } x \neq 1 \\ 1 & \quad \text{if } x = 1 \end{cases}\), \(a = 1\)
7 Continuity - Using Continuity · Level 2
Use continuity to evaluate the limit. \(\operatorname*{lim}\limits_{x \rightarrow 2} x \sqrt{20 - x^2}\)
8 Continuity - Domain/Theorems · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(F(x) = \sin(\cos(\sin x))\)
9 Continuity - Removable Discontinuity · Level 3
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 \(a\).
(a) \(f(x) = \dfrac{x^4 - 1}{x - 1}\), \(a = 1\)
(b) \(f(x) = \dfrac{x^3 - x^2 - 2x}{x - 2}\), \(a = 2\)
(c) \(f(x) = \lfloor \sin x \rfloor\), \(a = \pi\)

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10 Continuity - Applied · Level 2
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.
(a) Sketch a graph of \(T\) as a function of time \(t\), measured in hours past midnight.
(b) Discuss the discontinuities of this function and their significance.

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