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Section 2.3

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1. (a) \displaystyle{\lim_{x \to 2}(4x^2 - 3x) = 10}

(c) \displaystyle{\lim_{t \to 1} \frac{t^2 - 3}{t + 5} = -\frac{1}{3}}

2. (a) \displaystyle{\lim_{x \to 2} \frac{x^3 - 8}{x + 2} = 0}

(c) \displaystyle{\lim_{s \to -1} \frac{s^3 + 1}{s^4 - 1} = -\frac{3}{4}}

(e) \displaystyle{\lim_{t \to 2} \frac{t^3 - 8}{t -2} = 12}

(g) \displaystyle{\lim_{u \to 4} \frac{u}{(u - 4)^2} = \infty}

3. (a) \displaystyle{\lim_{x \to 1^+}(3x^2 + 4) = 7}

(c) \displaystyle{\lim_{x \to 3^+} \frac{1}{x - 3} = \infty}

(e) \displaystyle{\lim_{t \to -2^-} \frac{t}{t + 2} = \infty}

4. (a) \displaystyle{\lim_{x \to 2^+}\lfloor x \rfloor = 2}

(b) \displaystyle{\lim_{x \to 2^-}\lfloor x \rfloor = 1}

(c) \displaystyle{\lim_{x \to 3^-} \lceil x \rceil = 3}

(e) \displaystyle{\lim_{x \to 0^+}\lfloor \cos(x) \rfloor = 0}

5. (b) \displaystyle{\lim_{z \to 2^-}g(z) = 5}

(c) \displaystyle{\lim_{z \to 2^+}g(z) = 5}

(d) Yes: \displaystyle{\lim_{z \to 2}g(z) = 5}

7. (a) \displaystyle{\lim_{x \to \infty} (3x + 4) = \infty}

(b) \displaystyle{\lim_{x \to \infty} \frac{x^3 + 3x - 1}{2x^3 - x^2 + 21} = \frac{1}{2}}

(c) \displaystyle{\lim_{u \to \infty} \frac{u^4 + 3u - 6}{3u^2 + 1} = \infty}

(e) \displaystyle{\lim_{x \to -\infty} \frac{x^5 - 6x + 13}{x^2 + 18x - 25} = -\infty}

(g) \displaystyle{\lim_{v \to \infty} \sqrt{\frac{2v + 1}{v - 2}} = \sqrt{2}}

(i) \displaystyle{\lim_{x \to \infty} \frac{3x + 1}{\sqrt{4x^2 + 5}} = \frac{3}{2}}

(j) \displaystyle{\lim_{x \to -\infty} \frac{3x + 1}{\sqrt{4x^2 + 5}} = -\frac{3}{2}}

9. \displaystyle{\lim_{x \to \frac{\pi}{2}^-} \tan(x) = \infty}, \displaystyle{\lim_{x \to \frac{\pi}{2}^+} \tan(x) = -\infty}, \displaystyle{\lim_{x \to \frac{\pi}{2}} \tan(x)} does not exist

11. (a) Use the fact that -1 \le \sin(x) \le 1 for all x > 0 .

(b) \displaystyle{\lim_{x \to \infty}\frac{\sin(x)}{x} = 0}

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