# Exercises - Limits

1. Given the following,

$$\displaystyle{f(x) = \left\{ \begin{array}{ccc} \sqrt{16-x^2} &,& x \lt 4\\ x^2-10x+24 &,& 4 \lt x \lt 7\\ 4 &,& x \ge 7 \end{array} \right.}$$

Sketch $y=f(x)$, and then find each of the following:

1. The domain of $f$

2. The range of $f$

3. All $x$ and $y$ intercepts

4. $f\,(0)$

5. $f\,(7)$

6. $f\,(10)$

1. $f\,(5)$

2. $\displaystyle{\lim_{x \rightarrow 4} \ f\,(x)}$

3. $\displaystyle{\lim_{x \rightarrow 7} \ f\,(x)}$

4. $\displaystyle{\lim_{x \rightarrow -4} \ f\,(x)}$

2. Evaluate each limit given below, provided that it exists, and provide a graphical interpretation (hole, gap, vertical asymptote, point of continuity, etc...) for each related function at the $x$-value in question.

1. $\displaystyle{\lim_{x \rightarrow 1} \; (x^2-2)}$

2. $\displaystyle{\lim_{x \rightarrow -3} \; \frac{x^2 - 9}{x+3}}$

3. $\displaystyle{\lim_{x \rightarrow 2} \; \frac{x^2-4}{|x-2|}}$

4. $\displaystyle{\lim_{x \rightarrow -1} \; \frac{1}{(x+1)^2}}$

5. $\displaystyle{\lim_{x \rightarrow \pi/2} \; \cos^2(2x)}$   $\ans{\displaystyle{\lim_{x \rightarrow \pi/2} \cos^2 2x = \cos^2 \pi = (-1)^2 = 1}}$

6. $\displaystyle{\lim_{x \rightarrow 3} \; \frac{x-3}{x^2-9}}$

7. $\displaystyle{\lim_{x \rightarrow -3} \; \frac{x-3}{x^2-9}}$

8. $\displaystyle{\lim_{x \rightarrow 4} \; \frac{2-\sqrt{8-x}}{x-4}}$

1. $\displaystyle{\lim_{x \rightarrow 0} \; \frac{\sin 3x}{x}}$

2. $\displaystyle{\lim_{x \rightarrow 0} \; \frac{1-\cos x}{3x}}$

3. $\displaystyle{\lim_{x \rightarrow 2} \; (x^2 - 3x + 1)}$

4. $\displaystyle{\lim_{x \rightarrow -4} \; \frac{x^2-16}{x-4}}$   $\ans{\displaystyle{ \begin{array}{rcl} \displaystyle{\lim_{x \rightarrow 4}} \,\displaystyle{ \frac {\displaystyle{\frac{1}{\sqrt{x}} - \frac{1}{2} }}{x^2-16}} &=& \displaystyle{\lim_{x \rightarrow 4}} \,\displaystyle{ \frac {\displaystyle{\frac{2-\sqrt{x}}{2\sqrt{x}}}}{x^2-16}}\\\\ &=& \displaystyle{\lim_{x \rightarrow 4} \,\frac{2-x}{2\sqrt{x}(x^2-16)}}\\\\ &=& \displaystyle{\lim_{x \rightarrow 4} \,\frac{(2-\sqrt{x})(2+\sqrt{x})}{2\sqrt{x}(x^2-16)(2+\sqrt{x})}}\\\\ &=& \displaystyle{\lim_{x \rightarrow 4} \,\frac{4-x}{2\sqrt{x}(x^2-16)(2+\sqrt{x})}}\\\\ &=& \displaystyle{\lim_{x \rightarrow 4} \,\frac{x-4}{x-4} \cdot \frac{-1}{2\sqrt{x}(x+4)(2+\sqrt{x})}}\\\\ &=& \displaystyle{\lim_{x \rightarrow 4} \,\frac{-1}{2\sqrt{x}(x+4)(2+\sqrt{x})}}\\\\ &=& \displaystyle{\frac{-1}{2\sqrt{4}(4+4)(2+\sqrt{4})}}\\\\ &=& \displaystyle{\frac{-1}{(2)(2)(8)(4)}}\\\\ &=& \displaystyle{\frac{-1}{128}} \end{array}}}$

5. $\displaystyle{\lim_{x \rightarrow -4} \; \frac{x^2+3x-4}{x+4}}$

6. $\displaystyle{\lim_{x \rightarrow -3} \; \frac{x^2-9}{|x+3|}}$

7. $\displaystyle{\lim_{x \rightarrow 2} \; \frac{x^2+4}{x-2}}$

8. $\displaystyle{\lim_{x \rightarrow 3} \; \frac{x-5}{|x-3|}}$

3. Evaluate the given limit(s) provided they exist, and describe what happens in the graph of the function (hole, gap, vertical asymptote, point of continuity, etc...) at the related $x$-value(s).

1. $\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$;     $\displaystyle{f(x) = \left\{ \begin{array}{ccl} x+2 &,& x \lt 0\\ 0 &,& x=0\\ 2 &,& x \gt 0\\ \end{array} \right.}$

2. $\displaystyle{\lim_{x \rightarrow 4} \; f(x)}$;     $\displaystyle{f(x) = \left\{ \begin{array}{ccl} x+2 &,& x \neq 4\\ 2 &,& x = 4\\ \end{array}\right.}$

3. $\displaystyle{\lim_{x \rightarrow -2} \; f(x)}$;     $\displaystyle{f(x) = \left\{ \begin{array}{ccl} 5x+7 &,& x \lt -2\\ x^2 &,& x \gt -2\\ \end{array}\right.}$

4. $\displaystyle{\lim_{x \rightarrow 0} \; f\,(x)}$;     $\displaystyle{f(x) = \left\{ \begin{array}{ccc} x^2 &,& x \lt 0\\ 1+x &,& x \gt 0\\ \end{array}\right.}$

5. $\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$,   $\displaystyle{\lim_{x \rightarrow -1} \; f(x)}$;     $\displaystyle{f(x) = \left\{ \begin{array}{ccl} x+1 &,& x \lt 0\\ e^x &,& x \ge 0\\ \end{array} \right.}$

4. Evaluate the following limits, if they exist, and provide a graphical interpretation (hole, gap, vertical asymptote, point of continuity, etc...) of each related function.

1. $\displaystyle{\lim_{x \rightarrow 2} \; \frac{x^2-4}{x-2}}$

2. $\displaystyle{\lim_{x \rightarrow 0} \; \frac{\sqrt{x+3}-2}{x^2}}$

3. $\displaystyle{\lim_{x \rightarrow 1} \; \frac{x+1}{x^2-1}}$

4. $\displaystyle{\lim_{x \rightarrow 2} \; \frac{x-2}{x-2}}$

5. $\displaystyle{\lim_{x \rightarrow 4} \; \frac{1/\sqrt{x} - 1/2}{x^2-16}}$

6. $\displaystyle{\lim_{x \rightarrow 0} \; \frac{ 2 / (3x+1) - 3}{2x+3}}$

7. $\displaystyle{\lim_{x \rightarrow 8} \; \frac{\sqrt{x+1} + 2}{10}}$

8. $\displaystyle{\lim_{x \rightarrow 1} \; \frac{1 - 1/x^3}{x-1}}$

1. $\displaystyle{\lim_{x \rightarrow 7} \; \frac{3 - \sqrt{2x-5}}{x-7}}$

2. $\displaystyle{\lim_{x \rightarrow 2^{-}} \; \frac{x-2}{|x-2|}}$

3. $\displaystyle{\lim_{x \rightarrow 2} \; \frac{x^2-5x+6}{x^2-4x+4}}$

4. $\displaystyle{\lim_{x \rightarrow 5} \; \frac{x-5}{|x-3|}}$

5. $\displaystyle{\lim_{x \rightarrow 0} \; \frac{\sqrt{1+x} - 1}{2x}}$

6. $\displaystyle{\lim_{x \rightarrow 1} \; \sqrt{1-x}}$

7. $\displaystyle{\lim_{x \rightarrow 1} \; \frac{x^3 - x^2 - 2x + 2}{x^2 + x - 2}}$

8. $\displaystyle{\lim_{x \rightarrow 0} \left[ x \left( \frac{4}{x} -1 \right) \right]}$

5. Sketch the given function, and then find the indicated values.

1. $\displaystyle{g(x) = \left\{ \begin{array}{ccc} 10 &,& x \le -3\\ 1-3x &,& -3 \lt x \lt 1\\ 0 &,& x=1\\ x^2-3 &,& x \gt 1\\ \end{array}\right.}$

i) $\displaystyle{\lim_{x \rightarrow -4} \; g(x)}$       ii) $\displaystyle{\lim_{x \rightarrow -3} \; g(x)}$       iii) $\displaystyle{\lim_{x \rightarrow 0} \; g(x)}$       iv) $\displaystyle{\lim_{x \rightarrow 1} \; g(x)}$

2. $\displaystyle{f(x) = \left\{ \begin{array}{ccc} -\sqrt{16-x^2} &,& x \lt 0\\ 3 &,& x = 0\\ x^2 - 4 &,& 0 \lt x \le 3\\ 3 + \sqrt{7-x} &,& x \gt 3\\ \end{array}\right.}$

i) $\displaystyle{\lim_{x \rightarrow -4} \; f(x)}$       ii) $\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$       iii) $\displaystyle{\lim_{x \rightarrow 2} \; f(x)}$       iv) $\displaystyle{\lim_{x \rightarrow 3} \; f(x)}$       v) $f(0)$

3. $\displaystyle{f(x) = \left\{ \begin{array}{ccc} \sqrt{49-x^2} &,& x \le 0\\ 7 &,& 0 \lt x \le 3\\ x^2-8x+12 &,& x > 3\\ \end{array}\right.}$

i) $\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$       ii) $\displaystyle{\lim_{x \rightarrow 1} \; f(x)}$       iii) $\displaystyle{\lim_{x \rightarrow 3} \; f(x)}$       iv) $\displaystyle{\lim_{x \rightarrow 4} \; f(x)}$

4. $\displaystyle{f(x) = \left\{ \begin{array}{ccc} \sqrt{9-x^2} &,& x \lt 0\\ 3-x &,& 0 \le x \lt 3\\ 9-x^2 &,& x \gt 3 \end{array}\right.}$

i) $\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$       ii) $\displaystyle{\lim_{x \rightarrow 2} \; f(x)}$       iii) $\displaystyle{\lim_{x \rightarrow 3} \; f(x)}$

6. Evaluate the following limits and give a graphical interpretation of the behavior of the related function at the indicated $x$ value. When a limit below fails to exist, cite the reason why this is the case.

1. $\displaystyle{\lim_{x \rightarrow -6} \sqrt{4-2x}}$

2. $\displaystyle{\lim_{x \rightarrow 6} \frac{x^2-6x}{x^2-7x+6}}$

3. $\displaystyle{\lim_{x \rightarrow -4} \frac{x^2-16}{|4+x|}}$

4. $\displaystyle{\lim_{x \rightarrow 4} \frac{\sqrt{25-x^2}-3}{4-x}}$

5. $\displaystyle{\lim_{x \rightarrow -1} \frac{|x+1|}{x+1}}$

6. $\displaystyle{\lim_{x \rightarrow 0} \frac{\sin 4x}{2x}}$

7. $\displaystyle{\lim_{x \rightarrow 3} \frac{|x-3|}{x^2-9}}$

8. $\displaystyle{\lim_{x \rightarrow -2} \frac{1-\sqrt{x^2-3}}{4-x^2}}$

9. $\displaystyle{\lim_{x \rightarrow \pi} \tan x}$

10. $\displaystyle{\lim_{x \rightarrow 1} \frac{\frac{1}{x} - 1}{x-1}}$

11. $\displaystyle{\lim_{x \rightarrow 0} \frac{e^{2x}}{\ln(x+e)}}$

1. $\displaystyle{\lim_{x \rightarrow -3} \frac{x^2-4x+4}{x^2+x-6}}$

2. $\displaystyle{\lim_{x \rightarrow 1} \frac{x^3-8}{x-2}}$

3. $\displaystyle{\lim_{x \rightarrow -3} \frac{9-x^2}{x+3}}$

4. $\displaystyle{\lim_{x \rightarrow 5} \frac{x-5}{x+2}}$

5. $\displaystyle{\lim_{x \rightarrow \pi/4} \tan(2x)}$

6. $\displaystyle{\lim_{x \rightarrow 1} \textrm{ Arccos } \left( \frac{1}{x} \right)}$

7. $\displaystyle{\lim_{x \rightarrow 0} \frac{x}{1-\cos x}}$

8. $\displaystyle{\lim_{x \rightarrow -2} \frac{x^2-4}{x^3+8}}$

9. $\displaystyle{\lim_{x \rightarrow 1} \frac{\sqrt{2x+1} - \sqrt{3}}{x-1}}$

10. $\displaystyle{\lim_{x \rightarrow 3^{-}} \frac{|x-3|}{x-3}}$

11. $\displaystyle{\lim_{x \rightarrow 0} \frac{x}{e^x}}$

7. Evaluate the following limits and give a graphical interpretation of the behavior of the related function at the indicated $x$ value. When a limit below fails to exist, cite the reason why this is the case.

1. $\displaystyle{\lim_{x \rightarrow 0} \ e^{3x} \ln |1-x|}$

2. $\displaystyle{\lim_{x \rightarrow 2} \ \textrm{ Arcsin } \left( \frac{1}{x} \right)}$

3. $\displaystyle{\lim_{x \rightarrow \pi} \ \sec \left( \frac{x}{2} \right)}$

4. $\displaystyle{\lim_{x \rightarrow 4} \ \frac{\sqrt{25-x^2} - 3}{4-x}}$

5. $\displaystyle{\lim_{x \rightarrow -2} \ \frac{\frac{16}{x^4} - 1}{x^2-4}}$

6. $\displaystyle{\lim_{x \rightarrow 4} \ \frac{|x-4|}{x^2-16} }$

7. $\displaystyle{\lim_{x \rightarrow 0} \ \frac{\tan(2x)}{x}}$

8. $\displaystyle{\lim_{x \rightarrow 2^-} \ \tan \left(\frac{\pi}{x} \right)}$

9. $\displaystyle{\lim_{x \rightarrow 5} \ \frac{x^3-125}{x^2-25}}$

10. $\displaystyle{\lim_{x \rightarrow 0} \ \textrm{Arctan} (2x)}$

1. $\displaystyle{\lim_{x \rightarrow 0} \ \sin \left( \frac{1}{2x} \right)}$

2. $\displaystyle{\lim_{x \rightarrow 2} \ \textrm{Arctan} \left( \frac{x}{2} \right)}$

3. $\displaystyle{\lim_{x \rightarrow -2} \ \frac{|x+2|}{x^3+8}}$

4. $\displaystyle{\lim_{x \rightarrow -1} \ \textrm{Arcsin } \left( \frac{x}{2} \right)}$

5. $\displaystyle{\lim_{x \rightarrow 1} \ \frac{\frac{1}{x} - 1}{x^3 - 1}}$

6. $\displaystyle{\lim_{x \rightarrow -3} \ \textrm{Arctan}^2 \left( \frac{x}{3} \right)}$

7. $\displaystyle{\lim_{x \rightarrow 0} \ \sin \left( \frac{4}{3x} \right)}$

8. $\displaystyle{\lim_{x \rightarrow 4} \ \frac{|x-4|}{x^2-16}}$

9. $\displaystyle{\lim_{x \rightarrow -2} \ \frac{1-\sqrt{x^2-3}}{4-x^2}}$