Calculus Examples

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

Step 1

Apply trigonometric identities.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Divide $\cos (x - 1)$ by $1$ .

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

Step 1.3.2

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

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

Step 2

Consider the left sided limit.

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

Step 3

As the $x$ values approach $0$ from the left, the function values decrease without bound.

$- \infty$

Step 4

Consider the right sided limit.

$lim_{x \to 0^{+}} \cos (x - 1) \csc (x)$

Step 5

As the $x$ values approach $0$ from the right, the function values increase without bound.

$\infty$

Step 6

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$