Calculus Examples

$\frac{lim_{x \to 0} \ln (1 + 4 x)}{\sin (5 x)}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 0} 1 + 4 x)}{\sin (5 x)}$

Step 1.2

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

$\frac{\ln (lim_{x \to 0} 1 + lim_{x \to 0} 4 x)}{\sin (5 x)}$

Step 1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{\ln (1 + lim_{x \to 0} 4 x)}{\sin (5 x)}$

Step 1.4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\ln (1 + 4 lim_{x \to 0} x)}{\sin (5 x)}$

Step 2

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

$\frac{\ln (1 + 4 \cdot 0)}{\sin (5 x)}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Separate fractions.

$\frac{\ln (1 + 4 \cdot 0)}{1} \cdot \frac{1}{\sin (5 x)}$

Step 3.2

Convert from $\frac{1}{\sin (5 x)}$ to $\csc (5 x)$ .

$\frac{\ln (1 + 4 \cdot 0)}{1} \csc (5 x)$

Step 3.3

Multiply $4$ by $0$ .

$\frac{\ln (1 + 0)}{1} \csc (5 x)$

Step 3.4

Divide $\ln (1 + 0)$ by $1$ .

$\ln (1 + 0) \csc (5 x)$

Step 3.5

Add $1$ and $0$ .

$\ln (1) \csc (5 x)$

Step 3.6

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

$0 \csc (5 x)$

Step 3.7

Multiply $0$ by $\csc (5 x)$ .

$0$