Calculus Examples

$lim_{x \to 0} \frac{\ln (\sin (x))}{x - \frac{π}{2}}$

Step 1

Combine terms.

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Step 1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 0} \frac{\ln (\sin (x))}{x \cdot \frac{2}{2} - \frac{π}{2}}$

Step 1.2

Combine $x$ and $\frac{2}{2}$ .

$lim_{x \to 0} \frac{\ln (\sin (x))}{\frac{x \cdot 2}{2} - \frac{π}{2}}$

Step 1.3

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\ln (\sin (x))}{\frac{x \cdot 2 - π}{2}}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \ln (\sin (x)) \frac{2}{x \cdot 2 - π}$

Step 2.2

Combine $\ln (\sin (x))$ and $\frac{2}{x \cdot 2 - π}$ .

$lim_{x \to 0} \frac{\ln (\sin (x)) \cdot 2}{x \cdot 2 - π}$

Step 3

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} \frac{\ln (\sin (x)) \cdot 2}{x \cdot 2 - π}$

Step 4

Evaluate the limits by plugging in the value for the variable.

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Step 4.1

Evaluate the limit of $\frac{\ln (\sin (x)) \cdot 2}{x \cdot 2 - π}$ by plugging in $0$ for $x$ .

$\frac{\ln (\sin (0)) \cdot 2}{0 \cdot 2 - π}$

Step 4.2

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

$\frac{\ln (0) \cdot 2}{0 \cdot 2 - π}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Multiply $0$ by $2$ .

$\frac{\ln (0) \cdot 2}{0 - π}$

Step 4.3.2

Subtract $π$ from $0$ .

$\frac{\ln (0) \cdot 2}{- π}$

Step 4.3.3

The natural logarithm of zero is undefined.

Undefined

$\frac{\ln (0) \cdot 2}{- π}$

Step 4.4

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{\ln (0) \cdot 2}{- 1 (1) π}$

Step 4.5

Move the negative in front of the fraction.

$- \frac{\ln (0) \cdot 2}{(1) π}$

Step 4.6

Since $- \frac{\ln (0) \cdot 2}{(1) π}$ is undefined, the limit does not exist.

$Does not exist$

Step 5

Set up the limit as a right-sided limit.

$lim_{x \to 0^{+}} \frac{\ln (\sin (x)) \cdot 2}{x \cdot 2 - π}$

Step 6

Evaluate the right-sided limit.

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Step 6.1

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

$\frac{lim_{x \to 0^{+}} \ln (\sin (x)) \cdot 2}{lim_{x \to 0^{+}} x \cdot 2 - π}$

Step 6.2

Since the function $\ln (\sin (x))$ approaches $- \infty$ , the positive constant $2$ times the function also approaches $- \infty$ .

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Step 6.2.1

Consider the limit with the constant multiple $2$ removed.

$\frac{lim_{x \to 0^{+}} \ln (\sin (x))}{lim_{x \to 0^{+}} x \cdot 2 - π}$

Step 6.2.2

As $x$ approaches $0$ from the right side, $\ln (\sin (x))$ decreases without bound.

$\frac{- \infty}{lim_{x \to 0^{+}} x \cdot 2 - π}$

Step 6.3

Evaluate the limit.

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Step 6.3.1

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

$\frac{- \infty}{lim_{x \to 0^{+}} x \cdot 2 - lim_{x \to 0^{+}} π}$

Step 6.3.2

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

$\frac{- \infty}{2 lim_{x \to 0^{+}} x - lim_{x \to 0^{+}} π}$

Step 6.3.3

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

$\frac{- \infty}{2 lim_{x \to 0^{+}} x - π}$

Step 6.4

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

$\frac{- \infty}{2 \cdot 0 - π}$

Step 6.5

Simplify the answer.

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Step 6.5.1

Simplify the denominator.

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Step 6.5.1.1

Multiply $2$ by $0$ .

$\frac{- \infty}{0 - π}$

Step 6.5.1.2

Subtract $π$ from $0$ .

$\frac{- \infty}{- π}$

Step 6.5.2

Move the negative in front of the fraction.

$- \frac{- \infty}{π}$

Step 6.5.3

Infinity divided by anything that is finite and non-zero is infinity.

$- - \infty$

Step 6.5.4

A non-zero constant times infinity is infinity.

$\infty$

Step 7

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$