Calculus Examples

$\frac{lim_{x \to 0} 6 x \cos (4 x - 1)}{\sin (9 x - 9 x)}$

Step 1

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 lim_{x \to 0} x \cos (4 x - 1)}{\sin (9 x - 9 x)}$

Step 2

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

$\frac{6 lim_{x \to 0} x \cdot lim_{x \to 0} \cos (4 x - 1)}{\sin (9 x - 9 x)}$

Step 3

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

$\frac{6 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} 4 x - 1)}{\sin (9 x - 9 x)}$

Step 4

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

$\frac{6 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} 4 x - lim_{x \to 0} 1)}{\sin (9 x - 9 x)}$

Step 5

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

$\frac{6 lim_{x \to 0} x \cdot \cos (4 lim_{x \to 0} x - lim_{x \to 0} 1)}{\sin (9 x - 9 x)}$

Step 6

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

$\frac{6 lim_{x \to 0} x \cdot \cos (4 lim_{x \to 0} x - 1 \cdot 1)}{\sin (9 x - 9 x)}$

Step 7

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

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Step 7.1

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

$\frac{6 \cdot 0 \cdot \cos (4 lim_{x \to 0} x - 1 \cdot 1)}{\sin (9 x - 9 x)}$

Step 7.2

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

$\frac{6 \cdot 0 \cdot \cos (4 \cdot 0 - 1 \cdot 1)}{\sin (9 x - 9 x)}$

Step 8

Simplify the answer.

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Step 8.1

Subtract $9 x$ from $9 x$ .

$\frac{6 \cdot 0 \cdot \cos (4 \cdot 0 - 1 \cdot 1)}{\sin (0)}$

Step 8.2

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

$\frac{6 \cdot 0 \cdot \cos (4 \cdot 0 - 1 \cdot 1)}{0}$

Step 8.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{6 \cdot 0 \cdot \cos (4 \cdot 0 - 1 \cdot 1)}{0}$

Step 9

The expression contains a division by $0$ . The expression is undefined.

Undefined