Calculus Examples

$lim_{x \to 0} \frac{2 x^{2}}{\cos (x - 1)}$

Step 1

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

$2 lim_{x \to 0} \frac{x^{2}}{\cos (x - 1)}$

Step 2

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

$2 \frac{lim_{x \to 0} x^{2}}{lim_{x \to 0} \cos (x - 1)}$

Step 3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$2 \frac{{(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} \cos (x - 1)}$

Step 4

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

$2 \frac{{(lim_{x \to 0} x)}^{2}}{\cos (lim_{x \to 0} x - 1)}$

Step 5

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

$2 \frac{{(lim_{x \to 0} x)}^{2}}{\cos (lim_{x \to 0} x - lim_{x \to 0} 1)}$

Step 6

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

$2 \frac{{(lim_{x \to 0} x)}^{2}}{\cos (lim_{x \to 0} x - 1 \cdot 1)}$

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 $x$ .

$2 \frac{0^{2}}{\cos (lim_{x \to 0} x - 1 \cdot 1)}$

Step 7.2

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

$2 \frac{0^{2}}{\cos (0 - 1 \cdot 1)}$

Step 8

Simplify the answer.

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Step 8.1

Separate fractions.

$2 (\frac{0^{2}}{1} \cdot \frac{1}{\cos (0 - 1 \cdot 1)})$

Step 8.2

Convert from $\frac{1}{\cos (0 - 1 \cdot 1)}$ to $\sec (0 - 1 \cdot 1)$ .

$2 (\frac{0^{2}}{1} \sec (0 - 1 \cdot 1))$

Step 8.3

Multiply $- 1$ by $1$ .

$2 (\frac{0^{2}}{1} \sec (0 - 1))$

Step 8.4

Divide $0^{2}$ by $1$ .

$2 (0^{2} \sec (0 - 1))$

Step 8.5

Raising $0$ to any positive power yields $0$ .

$2 (0 \sec (0 - 1))$

Step 8.6

Subtract $1$ from $0$ .

$2 (0 \sec (- 1))$

Step 8.7

Evaluate $\sec (- 1)$ .

$2 (0 \cdot 1.00015232)$

Step 8.8

Multiply $2 (0 \cdot 1.00015232)$ .

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Step 8.8.1

Multiply $0$ by $1.00015232$ .

$2 \cdot 0$

Step 8.8.2

Multiply $2$ by $0$ .

$0$