Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Evaluate the limit of $\sinh (x)$ by plugging in $0$ for $x$ .

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

Step 6.3

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

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

Step 6.4

Evaluate the limit of $\cosh (x)$ by plugging in $0$ for $x$ .

$\frac{\sin (0) - \sinh (0)}{\cos (0) - \cosh (0)}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{0 - \sinh (0)}{\cos (0) - \cosh (0)}$

Step 7.1.2

Subtract $\sinh (0)$ from $0$ .

$\frac{- \sinh (0)}{\cos (0) - \cosh (0)}$

Step 7.2

The exact value of $\cos (0)$ is $1$ .

$\frac{- \sinh (0)}{1 - \cosh (0)}$

Step 7.3

Move the negative in front of the fraction.

$- \frac{\sinh (0)}{1 - \cosh (0)}$