Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Rewrite $\frac{\cos (0 - 1 \cdot 1)}{\sin (x)}$ as a product.

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

Step 3.2

Write $\cos (0 - 1 \cdot 1)$ as a fraction with denominator $1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Divide $\cos (0 - 1 \cdot 1)$ by $1$ .

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

Step 3.3.2

Multiply $- 1$ by $1$ .

$\cos (0 - 1) \frac{1}{\sin (x)}$

Step 3.3.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\cos (0 - 1) \csc (x)$

Step 3.4

Subtract $1$ from $0$ .

$\cos (- 1) \csc (x)$

Step 3.5

Evaluate $\cos (- 1)$ .

$0.99984769 \csc (x)$