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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 2.5 : Computing Limits
For problems 1 – 20 evaluate the limit, if it exists.
- \(\mathop {\lim }\limits_{x \to \, - 9} \left( {1 - 4{x^3}} \right)\)
- \(\mathop {\lim }\limits_{y \to 1} \left( {6{y^4} - 7{y^3} + 12y + 25} \right)\)
- \(\displaystyle \mathop {\lim }\limits_{t \to 0} \frac{{{t^2} + 6}}{{{t^2} - 3}}\)
- \(\displaystyle \mathop {\lim }\limits_{z \to 4} \frac{{6z}}{{2 + 3{z^2}}}\)
- \(\displaystyle \mathop {\lim }\limits_{w \to \, - 2} \frac{{w + 2}}{{{w^2} - 6w - 16}}\)
- \(\displaystyle \mathop {\lim }\limits_{t \to - 5} \frac{{{t^2} + 6t + 5}}{{{t^2} + 2t - 15}}\)
- \(\displaystyle \mathop {\lim }\limits_{x \to 3} \frac{{5{x^2} - 16x + 3}}{{9 - {x^2}}}\)
- \(\displaystyle \mathop {\lim }\limits_{z \to 1} \frac{{10 - 9z - {z^2}}}{{3{z^2} + 4z - 7}}\)
- \(\displaystyle \mathop {\lim }\limits_{x \to \, - 2} \frac{{{x^3} + 8}}{{{x^2} + 8x + 12}}\)
- \(\displaystyle \mathop {\lim }\limits_{t \to 8} \frac{{t\left( {t - 5} \right) - 24}}{{{t^2} - 8t}}\)
- \(\displaystyle \mathop {\lim }\limits_{w \to \, - 4} \frac{{{w^2} - 16}}{{\left( {w - 2} \right)\left( {w + 3} \right) - 6}}\)
- \(\displaystyle \mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {2 + h} \right)}^3} - 8}}{h}\)
- \(\displaystyle \mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {1 + h} \right)}^4} - 1}}{h}\)
- \(\displaystyle \mathop {\lim }\limits_{t \to 25} \frac{{5 - \sqrt t }}{{t - 25}}\)
- \(\displaystyle \mathop {\lim }\limits_{x \to \,2} \frac{{x - 2}}{{\sqrt 2 - \sqrt x }}\)
- \(\displaystyle \mathop {\lim }\limits_{z \to 6} \frac{{z - 6}}{{\sqrt {3z - 2} - 4}}\)
- \(\displaystyle \mathop {\lim }\limits_{z \to \, - 2} \frac{{3 - \sqrt {1 - 4z} }}{{2z + 4}}\)
- \(\displaystyle \mathop {\lim }\limits_{t \to 3} \frac{{3 - t}}{{\sqrt {t + 1} - \sqrt {5t - 11} }}\)
- \(\displaystyle \mathop {\lim }\limits_{x \to 7} \frac{{\,\frac{1}{7} - \frac{1}{x}\,}}{{x - 7}}\)
- \(\displaystyle \mathop {\lim }\limits_{y \to \, - 1} \frac{{\frac{1}{{4 + 3y}} + \frac{1}{y}}}{{y + 1}}\)
- Given the function
\[f\left( x \right) = \left\{ {\begin{align*}{15}&{\hspace{0.25in}x < - 4}\\{6 - 2x}&{\hspace{0.25in}x \ge - 4}\end{align*}} \right.\]
Evaluate the following limits, if they exist.
- \(\mathop {\lim }\limits_{x \to \, - 7} f\left( x \right)\)
- \(\mathop {\lim }\limits_{x \to - 4} f\left( x \right)\)
- Given the function
\[g\left( t \right) = \left\{ {\begin{align*}{{t^2} - {t^3}}&{\hspace{0.25in}t < 2}\\{5t - 14}&{\hspace{0.25in}t \ge 2}\end{align*}} \right.\]
Evaluate the following limits, if they exist.
- \(\mathop {\lim }\limits_{t \to - 3} g\left( t \right)\)
- \(\mathop {\lim }\limits_{t \to 2} g\left( t \right)\)
- Given the function
\[h\left( w \right) = \left\{ {\begin{align*}{2{w^2}}&{\hspace{0.25in}w \le 6}\\{w - 8}&{\hspace{0.25in}w > 6}\end{align*}} \right.\]
Evaluate the following limits, if they exist.
- \(\mathop {\lim }\limits_{w \to 6} h\left( w \right)\)
- \(\mathop {\lim }\limits_{w \to 2} h\left( w \right)\)
- Given the function
\[g\left( x \right) = \left\{ {\begin{array}{rc}{5x + 24}&{x < - 3}\\{{x^2}}&{ - 3 \le x < 4}\\{1 - 2x}&{x \ge 4}\end{array}} \right.\]
Evaluate the following limits, if they exist.
- \(\mathop {\lim }\limits_{x \to \, - 3} g\left( x \right)\)
- \(\mathop {\lim }\limits_{x \to \,0} g\left( x \right)\)
- \(\mathop {\lim }\limits_{x \to \,4} g\left( x \right)\)
- \(\mathop {\lim }\limits_{x \to \,12} g\left( x \right)\)
For problems 25 – 30 evaluate the limit, if it exists.
- \(\mathop {\lim }\limits_{z \to \, - 10} \left( {\left| {t + 10} \right| + 3} \right)\)
- \(\mathop {\lim }\limits_{x \to 4} \left( {9 + \left| {8 - 2x} \right|} \right)\)
- \(\displaystyle \mathop {\lim }\limits_{h \to 0} \frac{{\left| h \right|}}{h}\)
- \(\displaystyle \mathop {\lim }\limits_{t \to 2} \frac{{2 - t}}{{\left| {t - 2} \right|}}\)
- \(\displaystyle \mathop {\lim }\limits_{w \to \, - 5} \frac{{\left| {2w + 10} \right|}}{{w + 5}}\)
- \(\displaystyle \mathop {\lim }\limits_{x \to 4} \frac{{\left| {x - 4} \right|}}{{{x^2} - 16}}\)
- Given that \(3 + 2x \le f\left( x \right) \le x - 1\) for all x determine the value of \(\mathop {\lim }\limits_{x \to - 4} f\left( x \right)\).
- Given that \(\sqrt {x + 7} \le f\left( x \right) \le \frac{{x - 1}}{2}\) for all x determine the value of \(\mathop {\lim }\limits_{x \to 9} f\left( x \right)\).
- Use the Squeeze Theorem to determine the value of \(\displaystyle \mathop {\lim }\limits_{x \to 0} {x^4}\cos \left( {\frac{3}{x}} \right)\).
- Use the Squeeze Theorem to determine the value of \(\displaystyle \mathop {\lim }\limits_{x \to 0} x\cos \left( {\frac{1}{x}} \right)\).
- Use the Squeeze Theorem to determine the value of \(\displaystyle \mathop {\lim }\limits_{x \to 1} {\left( {x - 1} \right)^2}\cos \left( {\frac{1}{{x - 1}}} \right)\).