Calculus Examples

$lim_{x \to 4} \frac{1 - \cos (7 x)}{1 - \cos (4 x)}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 1 - \cos (7 x)}{lim_{x \to 4} 1 - \cos (4 x)}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 1 - lim_{x \to 4} \cos (7 x)}{lim_{x \to 4} 1 - \cos (4 x)}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$\frac{1 - lim_{x \to 4} \cos (7 x)}{lim_{x \to 4} 1 - \cos (4 x)}$

Step 4

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

$\frac{1 - \cos (lim_{x \to 4} 7 x)}{lim_{x \to 4} 1 - \cos (4 x)}$

Step 5

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 - \cos (7 lim_{x \to 4} x)}{lim_{x \to 4} 1 - \cos (4 x)}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{1 - \cos (7 lim_{x \to 4} x)}{lim_{x \to 4} 1 - lim_{x \to 4} \cos (4 x)}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$\frac{1 - \cos (7 lim_{x \to 4} x)}{1 - lim_{x \to 4} \cos (4 x)}$

Step 8

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

$\frac{1 - \cos (7 lim_{x \to 4} x)}{1 - \cos (lim_{x \to 4} 4 x)}$

Step 9

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

$\frac{1 - \cos (7 lim_{x \to 4} x)}{1 - \cos (4 lim_{x \to 4} x)}$

Step 10

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

Tap for more steps...

Step 10.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{1 - \cos (7 \cdot 4)}{1 - \cos (4 lim_{x \to 4} x)}$

Step 10.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{1 - \cos (7 \cdot 4)}{1 - \cos (4 \cdot 4)}$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1

Multiply $7$ by $4$ .

$\frac{1 - \cos (28)}{1 - \cos (4 \cdot 4)}$

Step 11.1.2

Evaluate $\cos (28)$ .

$\frac{1 - 1 \cdot 0.88294759}{1 - \cos (4 \cdot 4)}$

Step 11.1.3

Multiply $- 1$ by $0.88294759$ .

$\frac{1 - 0.88294759}{1 - \cos (4 \cdot 4)}$

Step 11.1.4

Subtract $0.88294759$ from $1$ .

$\frac{0.1170524}{1 - \cos (4 \cdot 4)}$

Step 11.2

Simplify the denominator.

Tap for more steps...

Step 11.2.1

Multiply $4$ by $4$ .

$\frac{0.1170524}{1 - \cos (16)}$

Step 11.2.2

Evaluate $\cos (16)$ .

$\frac{0.1170524}{1 - 1 \cdot 0.96126169}$

Step 11.2.3

Multiply $- 1$ by $0.96126169$ .

$\frac{0.1170524}{1 - 0.96126169}$

Step 11.2.4

Subtract $0.96126169$ from $1$ .

$\frac{0.1170524}{0.0387383}$

Step 11.3

Divide $0.1170524$ by $0.0387383$ .

$3.02161929$