Calculus Examples

$lim_{x \to 1} \ln (\frac{x}{e^{2 x}})$

Step 1

Move the limit inside the logarithm.

$\ln (lim_{x \to 1} \frac{x}{e^{2 x}})$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\ln (\frac{lim_{x \to 1} x}{lim_{x \to 1} e^{2 x}})$

Step 3

Move the limit into the exponent.

$\ln (\frac{lim_{x \to 1} x}{e^{lim_{x \to 1} 2 x}})$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\ln (\frac{lim_{x \to 1} x}{e^{2 lim_{x \to 1} x}})$

Step 5

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\ln (\frac{1}{e^{2 lim_{x \to 1} x}})$

Step 5.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\ln (\frac{1}{e^{2 \cdot 1}})$

Step 6

Simplify the answer.

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Step 6.1

Move $e^{2 \cdot 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln (e^{- 2})$

Step 6.2

Expand $\ln (e^{- 2})$ by moving $- 2$ outside the logarithm.

$- 2 \ln (e)$

Step 6.3

The natural logarithm of $e$ is $1$ .

$- 2 \cdot 1$

Step 6.4

Multiply $- 2$ by $1$ .

$- 2$