« Section 5.8
Section 6.1 »

Section 5.9

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1. (a) \displaystyle{\lim_{x \to 0} \frac{\sin(3x)}{x} = 3}

(c) \displaystyle{\lim_{x \to 0} \frac{\cos(x) - 1 + \dfrac{x^2}{2}}{x^4} = \frac{1}{24}}

(e) \displaystyle{\lim_{u \to 0} \frac{\tan(u)}{\sin(u)} = 1}

(g) \displaystyle{\lim_{y \to 0} \frac{\tan(3y)}{\tan(5y)} = \frac{3}{5}}

(i) \displaystyle{\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{3x} = \frac{1}{6}}

2. (a) \displaystyle{\lim_{x \to 0} \frac{\sin(5x)}{3x} = \frac{5}{3}}

(c) \displaystyle{\lim_{x \to 0} \frac{1 - \sec(x)}{x} = 0}

(e) \displaystyle{\lim_{x \to 0^+} \frac{\sin(2x)}{\sqrt{x}} = 0}

(g) \displaystyle{\lim_{x \to \infty}x\sin\left(\frac{1}{x^2}\right) = 0}

3.(a) \displaystyle{\lim_{x \to \infty} \frac{3x^2 - 2x + 1}{16x^2 + 2} = \frac{3}{16}}

(c) \displaystyle{\lim_{x \to 0}\frac{1 - \dfrac{\sin(x)}{x}}{3x^2} = \frac{1}{18}}

(e) \displaystyle{\lim_{x \to 0} \frac{(1+x)^{\frac{1}{3}} - 1 - \dfrac{x}{3}}{x^2} = -\frac{1}{9}}

(g) \displaystyle{\lim_{x \to 1}\frac{\sqrt{x} - 1}{x - 1} = \frac{1}{2}}

(i) \displaystyle{\lim_{t \to 0}\frac{\cos(t) - 1 + \dfrac{t^2}{2}}{t^2\sin(t^2)} = \frac{1}{24}}

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