Calculus Examples

$lim_{x \to π} \frac{\ln (1 + \sin (x))}{x}$

Step 1

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

$\frac{lim_{x \to π} \ln (1 + \sin (x))}{lim_{x \to π} x}$

Step 2

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to π} 1 + \sin (x))}{lim_{x \to π} x}$

Step 3

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

$\frac{\ln (lim_{x \to π} 1 + lim_{x \to π} \sin (x))}{lim_{x \to π} x}$

Step 4

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

$\frac{\ln (1 + lim_{x \to π} \sin (x))}{lim_{x \to π} x}$

Step 5

Move the limit inside the trig function because sine is continuous.

$\frac{\ln (1 + \sin (lim_{x \to π} x))}{lim_{x \to π} x}$

Step 6

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

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Step 6.1

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

$\frac{\ln (1 + \sin (π))}{lim_{x \to π} x}$

Step 6.2

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

$\frac{\ln (1 + \sin (π))}{π}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\ln (1 + \sin (0))}{π}$

Step 7.1.1.2

The exact value of $\sin (0)$ is $0$ .

$\frac{\ln (1 + 0)}{π}$

Step 7.1.2

Add $1$ and $0$ .

$\frac{\ln (1)}{π}$

Step 7.1.3

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

$\frac{0}{π}$

Step 7.2

Divide $0$ by $π$ .

$0$