Calculus Examples

$\frac{lim_{x \to 0} \ln (x + e^{x})}{e^{\sin (x)} - \cos (x)}$

Step 1

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 0} x + e^{x})}{e^{\sin (x)} - \cos (x)}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\ln (lim_{x \to 0} x + lim_{x \to 0} e^{x})}{e^{\sin (x)} - \cos (x)}$

Step 3

Move the limit into the exponent.

$\frac{\ln (lim_{x \to 0} x + e^{lim_{x \to 0} x})}{e^{\sin (x)} - \cos (x)}$

Step 4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\ln (0 + e^{lim_{x \to 0} x})}{e^{\sin (x)} - \cos (x)}$

Step 4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\ln (0 + e^{0})}{e^{\sin (x)} - \cos (x)}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Anything raised to $0$ is $1$ .

$\frac{\ln (0 + 1)}{e^{\sin (x)} - \cos (x)}$

Step 5.1.2

Add $0$ and $1$ .

$\frac{\ln (1)}{e^{\sin (x)} - \cos (x)}$

Step 5.1.3

The natural logarithm of $1$ is $0$ .

$\frac{0}{e^{\sin (x)} - \cos (x)}$

Step 5.2

Divide $0$ by $e^{\sin (x)} - \cos (x)$ .

$0$