Calculus Examples

$\frac{lim_{x \to 0} \ln (\cos (2 x))}{9 x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 0} \cos (2 x))}{9 x^{2}}$

Step 1.2

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

$\frac{\ln (\cos (lim_{x \to 0} 2 x))}{9 x^{2}}$

Step 1.3

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

$\frac{\ln (\cos (2 lim_{x \to 0} x))}{9 x^{2}}$

Step 2

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

$\frac{\ln (\cos (2 \cdot 0))}{9 x^{2}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $2$ by $0$ .

$\frac{\ln (\cos (0))}{9 x^{2}}$

Step 3.1.2

The exact value of $\cos (0)$ is $1$ .

$\frac{\ln (1)}{9 x^{2}}$

Step 3.1.3

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

$\frac{0}{9 x^{2}}$

Step 3.2

Cancel the common factor of $0$ and $9$ .

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Step 3.2.1

Factor $9$ out of $0$ .

$\frac{9 (0)}{9 x^{2}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $9$ out of $9 x^{2}$ .

$\frac{9 (0)}{9 (x^{2})}$

Step 3.2.2.2

Cancel the common factor.

$\frac{9 \cdot 0}{9 x^{2}}$

Step 3.2.2.3

Rewrite the expression.

$\frac{0}{x^{2}}$

Step 3.3

Divide $0$ by $x^{2}$ .

$0$