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