Stewart Precalc 6e Chapter 13 Test

1 Limit - Table of Values · Level 2

(a) Use a table of values to estimate the limit \(\operatorname*{lim}\limits_{x \rightarrow 0} \dfrac{x}{\sin}(2 x)\)

Answer for 1(a)

(b) Use a graphing calculator to confirm your answer graphically.

Answer for 1(b)

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2 Limits from Graph · Level 2

For the piecewise-defined function \(f\) whose graph is shown, find:

(a) \(\operatorname*{lim}\limits_{x \rightarrow -1^-} f(x)\)

Answer for 2(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow -1^+} f(x)\)

Answer for 2(b)

(c) \(\operatorname*{lim}\limits_{x \rightarrow -1} f(x)\)

Answer for 2(c)

(d) \(\operatorname*{lim}\limits_{x \rightarrow 0^-} f(x)\)

Answer for 2(d)

(e) \(\operatorname*{lim}\limits_{x \rightarrow 0^+} f(x)\)

Answer for 2(e)

(f) \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x)\) (g) \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x)\) (h) \(\operatorname*{lim}\limits_{x \rightarrow 4^-} f(x)\) (i) \(\operatorname*{lim}\limits_{x \rightarrow 4^+} f(x)\)

Answer for 2(f)

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3 Evaluate Limits · Level 2

Evaluate the limit if it exists.

(a) \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 + 2 x - 8}{x - 2}\)

Answer for 3(a)

(b) \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{x^2 - 2 x - 8}{x + 2}\)

Answer for 3(b)

(c) \(\operatorname*{lim}\limits_{x \rightarrow 2} 1/(x - 2)\)

Answer for 3(c)

(d) \(\operatorname*{lim}\limits_{x \rightarrow 2} (x - 2)/|x - 2|\)

Answer for 3(d)

(e) \(\operatorname*{lim}\limits_{x \rightarrow 4} \dfrac{\sqrt{x} - 2}{x - 4}\)

Answer for 3(e)

(f) \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{2 x^2 - 4}{x^2 + x}\)

Answer for 3(f)

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4 Derivative · Level 2

Let \(f(x) = x^2 - 2 x\). Find:

(a) \(f'(x)\)

Answer for 4(a)

(b) \(f'(-1), f'(1), f'(2)\)

Answer for 4(b)

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5 Tangent Line · Level 2

Find the equation of the line tangent to the graph of \(f(x) = \sqrt{x}\) at the point where \(x = 4\).

6 Sequence Limit · Level 2

Find the limit of the sequence.

(a) \(a_n = n/(n^2 + 4)\)

Answer for 6(a)

(b) \(a_n = \sec(n \pi)\)

Answer for 6(b)

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7 Area Under Curve · Level 3

The region sketched in the figure in the margin lies under the graph of \(f(x) = 4 - x^2\), above the interval \(0 \leq x \leq 1\).

(a) Approximate the area of the region with five rectangles, equally spaced along the \(x\)-axis, using right endpoints to determine the heights of the rectangles.

Answer for 7(a)

(b) Use the limit definition of area to find the exact value of the area of the region.

Answer for 7(b)

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