Quick Overview
- Basic form: $$\displaystyle \lim_{u\to0}\frac{e^u-1} u = 1$$
- Note that the denominator must match the exponent and that both must be going to zero in the limit .
Examples
Example 1
Evaluate $$\displaystyle \lim_{x\to0}\,\frac{e^{4x}-1} x$$
Multiply by $$\frac 4 4$$
$$ \begin{align*} \displaystyle\lim_{x\to0}\,\frac{e^{4x}-1} x % & = \displaystyle\lim_{x\to0}\left(\frac{\blue 4}{\blue 4}\cdot\frac{e^{4x}-1} x\right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(\frac{\blue 4}{1}\cdot\frac{e^{4x}-1} {\blue 4x}\right)\\[6pt] % & = \blue 4 \cdot \lim_{x\to0}\frac{e^{4x}-1} {4x} \end{align*} $$
Evaluate the limit .
$$ 4 \cdot \blue{\lim_{x\to0}\frac{e^{4x}-1} {4x}} % = 4(\blue 1) % =4 $$
$$\displaystyle \lim_{x\to0}\,\frac{e^{4x}-1} x = 4$$
Example 2
Evaluate: $$\displaystyle \lim_{x\to0}\, \frac x {e^{10x} - 1}$$
Multiply by $$\frac{10}{10}$$
$$ \\ \begin{align*} \displaystyle\lim_{x\to0}\,\frac x {e^{10x} - 1} % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{10}}{\blue{10}}\cdot \frac x {e^{10x} - 1} \right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(% \frac{1}{\blue{10}}\cdot \frac{\blue{10}x}{e^{10x} - 1} \right)\\[6pt] % & = \frac{1}{\blue{10}}\cdot \lim_{x\to0}\,\frac{\blue{10}x}{e^{10x} - 1} \end{align*} \\ $$
Rewrite the function as its reciprocal raised to the $$-1$$ power.
$$ \frac{1}{10}\cdot \displaystyle\lim_{x\to0}\,\frac{10x}{e^{10x} - 1} % = \frac{1}{10}\cdot \displaystyle\lim_{x\to0}\left(% \frac{e^{10x} - 1}{10x} \right)^{-1} $$
Pass the limit inside the exponent and evaluate (see the page on Limit Laws).
$$ \frac{1}{10}\cdot \displaystyle\lim_{x\to0}\left(% \blue{\frac{e^{10x} - 1}{10x}} \right)^{-1} % = \frac{1}{10}\left(% \blue{\displaystyle\lim_{x\to0}\,\frac{e^{10x} - 1}{10x}} \right)^{-1} % = \frac 1 {10} (\blue 1)^{-1} % = \frac 1 {10} $$
$$\\ \displaystyle \lim_{x\to0}\, \frac x {e^{10x} - 1} = \frac 1 {10} \\$$
Example 3
Evaluate: $$\displaystyle \lim_{x\to0}\, \frac{e^{2x}-1}{e^{7x}-1}$$
Rewrite in separate fractions.
$$ \displaystyle\lim_{x\to0}\, \frac{e^{2x}-1}{e^{7x}-1} % = \displaystyle\lim_{x\to0}\left( \frac{e^{2x}-1} 1 \cdot \frac 1 {e^{7x}-1}\right) $$
Multiply by $$\frac{2x}{2x}$$ and $$\frac{7x}{7x}$$
$$ \begin{align*} \displaystyle\lim_{x\to0}\left( \frac{e^{2x}-1} 1 \cdot \frac 1 {e^{7x}-1}\right) % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{2x}}{\blue{2x}}\cdot \frac{e^{2x}-1} 1 \cdot \frac{\red{7x}}{\red{7x}}\cdot \frac 1 {e^{7x}-1} \right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{2x}} 1\cdot \frac{e^{2x}-1}{\blue{2x}}\cdot \frac 1 {\red{7x}}\cdot \frac{\red{7x}}{e^{7x}-1} \right) \end{align*} $$
Simplify the non-exponential fractions.
$$ \begin{align*} \displaystyle\lim_{x\to0}\left(% \frac{\blue{2x}}{\blue 1}\cdot \frac{e^{2x}-1}{2x}\cdot \frac{\red 1}{\red{7x}}\cdot \frac{7x}{e^{7x}-1} \right) % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{2x}}{\red{7x}}\cdot \frac{e^{2x}-1}{2x}\cdot \frac{7x}{e^{7x}-1} \right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{2}}{\red{7}}\cdot \frac{e^{2x}-1}{2x}\cdot \frac{7x}{e^{7x}-1} \right)\\[6pt] % & = \frac{\blue{2}}{\red{7}}\cdot\displaystyle\lim_{x\to0}\left(% \frac{e^{2x}-1}{2x}\cdot \frac{7x}{e^{7x}-1} \right) \end{align*} $$
Evaluate the limit of each factor.
$$ \frac 2 7\cdot\displaystyle\lim_{x\to0}\left(% \blue{\frac{e^{2x}-1}{2x}}\cdot \red{\frac{7x}{e^{7x}-1}} \right) % = \frac 2 7 (\blue 1)(\red 1) % = \frac 2 7 $$
$$\\ \displaystyle \lim_{x\to0}\, \frac{e^{2x}-1}{e^{7x}-1} = \frac 2 7 \\$$
Example 4
Evaluate: $$\displaystyle \lim_{x\to0}\,\frac{e^{8x}-1}{\sin 6x}$$
Rewrite in separate fractions.
$$ \displaystyle\lim_{x\to0}\,\frac{e^{8x}-1}{\sin 6x} % = \displaystyle\lim_{x\to0}\left(% \frac{e^{8x}-1} 1 \cdot \frac 1 {\sin 6x} \right) $$
Multiply by $$\frac{8x}{8x}$$ and $$\frac{6x}{6x}$$
$$ \begin{align*} \lim_{x\to0}\left(\frac{e^{8x}-1} 1 \cdot \frac 1 {\sin 6x}\right) % & = \lim_{x\to0}\left(% \frac{\blue{8x}}{\blue{8x}}\cdot \frac{e^{8x}-1} 1 \cdot \frac{\red{6x}}{\red{6x}}\cdot \frac 1 {\sin 6x} \right)\\[6pt] % & = \lim_{x\to0}\left(% \frac{\blue{8x}} 1\cdot \frac{e^{8x}-1}{\blue{8x}}\cdot \frac 1 {\red{6x}}\cdot \frac{\red{6x}}{\sin 6x} \right) \end{align*} $$
Simplify the non-transcendental fractions.
$$ \begin{align*} \lim_{x\to0}\left(% \blue{\frac{8x} 1}\cdot \frac{e^{8x}-1}{8x}\cdot \red{\frac 1 {6x}}\cdot \frac{6x}{\sin 6x} \right) % & = \lim_{x\to0}\left(% \frac{\blue{8x}}{\red{6x}}\cdot \frac{e^{8x}-1}{8x}\cdot \frac{6x}{\sin 6x} \right)\\[6pt] % & = \lim_{x\to0}\left(% \frac{\blue{4}}{\red{3}}\cdot \frac{e^{8x}-1}{8x}\cdot \frac{6x}{\sin 6x} \right)\\[6pt] % & = \frac{\blue{4}}{\red{3}}\cdot\lim_{x\to0}\left(% \frac{e^{8x}-1}{8x}\cdot \frac{6x}{\sin 6x} \right) \end{align*} $$
Evaluate the limit of each factor.
$$ \frac 4 3\cdot\displaystyle\lim_{x\to0}\left(% \blue{\frac{e^{8x}-1}{8x}}\cdot \red{\frac{6x}{\sin 6x}} \right) % = \frac 4 3 (\blue 1)(\red 1) % = \frac 4 3 $$
$$\displaystyle \lim_{x\to0}\,\frac{e^{8x}-1}{\sin 6x} = \frac 4 3$$
Practice Problems
Multiply by $$\frac{-4}{-4}$$.
$$ \begin{align*} \displaystyle\lim_{x\to0}\,\frac{e^{-4x} - 1} x % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{-4}}{\blue{-4}}\cdot \frac{e^{-4x} - 1} x \right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{-4}}{1}\cdot \frac{e^{-4x} - 1}{\blue{-4}x} \right)\\[6pt] % & = \blue{-4}\cdot\displaystyle\lim_{x\to0}\,% \frac{e^{-4x} - 1}{\blue{-4}x} \end{align*} $$
Evaluate the limit .
$$ -4\cdot\blue{\displaystyle\lim_{x\to0}\,% \frac{e^{-4x} - 1}{-4x}} % = -4(\blue 1) % = -4 $$
$$\displaystyle \lim_{x\to0}\,\frac{e^{-4x} - 1} x = -4$$.
Factor the 21 out of the denominator.
$$ \displaystyle\lim_{x\to0}\,\frac{e^{6x} - 1}{\blue{21}x} % = \displaystyle\lim_{x\to0}\left(% \frac 1 {\blue{21}}\cdot \frac{e^{6x} - 1}{x} \right) % = \frac 1 {\blue{21}}\cdot \displaystyle\lim_{x\to0}\,\frac{e^{6x}-1} x $$
Multiply by $$\frac 6 6$$.
$$ \begin{align*} \frac 1 {21}\cdot \lim_{x\to0}\,\frac{e^{6x}-1} x % & = \frac 1 {21} \cdot\lim_{x\to0}\left(% \frac{\blue 6}{\blue 6}\cdot \frac{e^{6x}-1} x \right)\\[6pt] % & = \frac 1 {21} \cdot\lim_{x\to0}\left(% \frac{\blue 6} 1 \cdot \frac{e^{6x}-1} {{\blue 6}x} \right)\\[6pt] % & = \frac{\blue 6}{21}\cdot \lim_{x\to0}\,% \frac{e^{6x}-1} {{\blue 6}x}\\[6pt] % & = \frac 2 7\cdot \lim_{x\to0}\,% \frac{e^{6x}-1} {6x} \end{align*} $$
Evaluate the limit .
$$ \frac 2 7\cdot \blue{\displaystyle\lim_{x\to0}\,% \frac{e^{6x}-1} {6x}} % = \frac 2 7 (\blue 1) % = \frac 2 7 $$
$$\displaystyle \lim_{x\to0}\,\frac{e^{6x} - 1}{21x} = \frac 2 7$$.
Factor the 10 out of the numerator.
$$ \displaystyle\lim_{x\to0}\,\frac{\blue{10}x}{e^{8x} - 1} % = \displaystyle\lim_{x\to0}\left(% \blue{10}\cdot\frac{x}{e^{8x} - 1} \right) % = \blue{10}\cdot\displaystyle\lim_{x\to0}\,\frac{x}{e^{8x} - 1} $$
Multiply by $$\frac 8 8$$
$$ \begin{align*} 10\cdot\lim_{x\to0}\,\frac{x}{e^{8x} - 1} % & = 10\cdot\lim_{x\to0}\left(% \frac{\blue 8}{\blue 8}\cdot \frac{x}{e^{8x} - 1} \right)\\[6pt] % & = 10\cdot\lim_{x\to0}\left(% \frac{1}{\blue 8}\cdot \frac{\blue 8 x}{e^{8x} - 1} \right)\\[6pt] % & = \frac{10}{\blue 8}\cdot\lim_{x\to0}\,% \frac{\blue 8 x}{e^{8x} - 1}\\[6pt] % & = \frac 5 4\cdot\lim_{x\to0}\,% \frac{8x}{e^{8x} - 1} \end{align*} $$
Evaluate the limit .
$$ \frac 5 4\cdot\blue{\displaystyle\lim_{x\to0}\,% \frac{8x}{e^{8x} - 1}} % = \frac 5 4 (\blue 1) % = \frac 5 4 $$
$$\displaystyle \lim_{x\to0}\, \frac{10x}{e^{8x}-1} = \frac 5 4$$.
Factor out the 2 from the numerator.
$$ \displaystyle\lim_{x\to0}\,\frac{\blue 2 x}{e^{-7x} - 1} % = \displaystyle\lim_{x\to0}\left(% \blue 2\cdot \frac{x}{e^{-7x} - 1} \right) % = \blue 2\cdot \displaystyle\lim_{x\to0}\,% \frac{x}{e^{-7x} - 1} $$
Multiply by $$\frac{-7}{-7}$$
$$ \begin{align*} 2\cdot \displaystyle\lim_{x\to0}\,\frac{x}{e^{-7x} - 1} % & = 2\cdot \displaystyle\lim_{x\to0}\left(% \frac{\blue{-7}}{\blue{-7}}\cdot \frac{x}{e^{-7x} - 1} \right)\\[6pt] % & = 2\cdot \displaystyle\lim_{x\to0}\left(% \frac{1}{\blue{-7}}\cdot \frac{\blue{-7}x}{e^{-7x} - 1} \right)\\[6pt] % & = -\frac 2 {\blue 7}\cdot \displaystyle\lim_{x\to0}\,% \frac{-7x}{e^{-7x} - 1} \end{align*} $$
Evaluate the limit .
$$ -\frac 2 7\cdot \blue{\displaystyle\lim_{x\to0}\,% \frac{-7x}{e^{-7x} - 1}} % = -\frac 2 7 (\blue 1) % = -\frac 2 7 $$.
$$ \displaystyle \lim_{x\to0}\,\frac{2 x}{e^{-7x} - 1} = -\frac 2 7 $$.
Rewrite the numerator and denominator in separate fractions.
$$ \displaystyle\lim_{x\to0}\,\frac{e^{5x}-1}{e^{6x}-1} % = \displaystyle\lim_{x\to0}\left(% \frac{e^{5x}-1} 1\cdot \frac 1 {e^{6x}-1} \right) $$
Multiply by $$\frac{5x}{5x}$$ and $$\frac{6x}{6x}$$
$$ \begin{align*} \lim_{x\to0}\left(% \frac{e^{5x}-1} 1\cdot \frac 1 {e^{6x}-1} \right) % & = \lim_{x\to0}\left(% \frac{\blue {5x}}{\blue{5x}}\cdot \frac{e^{5x}-1} 1\cdot \frac{\red{6x}}{\red{6x}}\cdot \frac 1 {e^{6x}-1} \right)\\[6pt] % & = \lim_{x\to0}\left(% \frac{\blue {5x}}{1}\cdot \frac{e^{5x}-1}{\blue{5x}}\cdot \frac{1}{\red{6x}}\cdot \frac{\red{6x}}{e^{6x}-1} \right) \end{align*} $$
Simplify the non-exponential fractions.
$$ \begin{align*} \displaystyle\lim_{x\to0}\left(% \frac{\blue {5x}}{\blue 1}\cdot \frac{e^{5x}-1}{5x}\cdot \frac{\red 1}{\red{6x}}\cdot \frac{6x}{e^{6x}-1} \right) % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue {5x}}{\red{6x}}\cdot \frac{e^{5x}-1}{5x}\cdot \frac{6x}{e^{6x}-1} \right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue {5}}{\red{6}}\cdot \frac{e^{5x}-1}{5x}\cdot \frac{6x}{e^{6x}-1} \right)\\[6pt] % & = \frac{\blue {5}}{\red{6}}\cdot\displaystyle\lim_{x\to0}\left(% \frac{e^{5x}-1}{5x}\cdot \frac{6x}{e^{6x}-1} \right) \end{align*} $$
Evaluate the limit of each factor.
$$ \frac 5 6\cdot\displaystyle\lim_{x\to0}\left(% \blue{\frac{e^{5x}-1}{5x}}\cdot \red{\frac{6x}{e^{6x}-1}} \right) % = \frac 5 6 (\blue 1)(\red 1) % = \frac 5 6 $$
$$ \displaystyle \lim_{x\to0}\,\frac{e^{5x}-1}{e^{6x}-1} = \frac 5 6 $$
Write the numerator and denominator in separate fractions.
$$ \displaystyle\lim_{x\to0}\,\frac{1-e^{x/2}}{e^{3x}-1} % = \displaystyle\lim_{x\to0}\left(% \frac{1-e^{x/2}} 1\cdot \frac 1 {e^{3x}-1} \right) $$
Multiply by $$\frac{x/2}{x/2}$$ and $$\frac{3x}{3x}$$.
$$ \begin{align*} \displaystyle\lim_{x\to0}\left(% \frac{1-e^{x/2}} 1\cdot \frac 1 {e^{3x}-1} \right) % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{x/2}}{\blue{x/2}}\cdot \frac{1-e^{x/2}} 1\cdot \frac{\red{3x}}{\red{3x}}\cdot \frac 1 {e^{3x}-1} \right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue{x/2}} 1\cdot \frac{1-e^{x/2}}{\blue{x/2}}\cdot \frac 1 {\red{3x}}\cdot \frac{\red{3x}}{e^{3x}-1} \right) \end{align*} $$
Simplify the non-exponential fractions.
$$ \begin{align*} \displaystyle\lim_{x\to0}\left(% \frac{\blue{x/2}}{\blue 1}\cdot \frac{1-e^{x/2}}{x/2}\cdot \frac{\red 1}{\red{3x}}\cdot \frac{3x}{e^{3x}-1} \right) % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue x}{\blue 2}\cdot \frac{1-e^{x/2}}{x/2}\cdot \frac{\red 1}{\red{3x}}\cdot \frac{3x}{e^{3x}-1} \right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue x}{\blue{6x}}\cdot \frac{1-e^{x/2}}{x/2}\cdot \frac{3x}{e^{3x}-1} \right)\\[6pt] % & = \displaystyle\lim_{x\to0}\left(% \frac{\blue 1}{\blue 6}\cdot \frac{1-e^{x/2}}{x/2}\cdot \frac{3x}{e^{3x}-1} \right)\\[6pt] % & = \displaystyle\frac{\blue 1}{\blue 6}\cdot\displaystyle\lim_{x\to0}\left(% \frac{1-e^{x/2}}{x/2}\cdot \frac{3x}{e^{3x}-1} \right) \end{align*} $$.
Evaluate the limit of each factor.
$$ \frac 1 6\cdot\displaystyle\lim_{x\to0}\left(% \blue{\frac{1-e^{x/2}}{x/2}}\cdot \red{\frac{3x}{e^{3x}-1}} \right) % = \frac 1 6 (\blue 1)(\red 1) % = \frac 1 6 $$
$$ \displaystyle \lim_{x\to0}\,\frac{1-e^{x/2}}{e^{3x}-1} = \frac 1 6 $$
