Calculus Examples

$lim_{x \to \frac{π}{4}} \frac{\sin (x) - \cos (x)}{\tan (x - 1)}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}{lim_{x \to \frac{π}{4}} \tan (x - 1)}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{lim_{x \to \frac{π}{4}} \sin (x) - lim_{x \to \frac{π}{4}} \cos (x)}{lim_{x \to \frac{π}{4}} \tan (x - 1)}$

Step 3

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

$\frac{\sin (lim_{x \to \frac{π}{4}} x) - lim_{x \to \frac{π}{4}} \cos (x)}{lim_{x \to \frac{π}{4}} \tan (x - 1)}$

Step 4

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

$\frac{\sin (lim_{x \to \frac{π}{4}} x) - \cos (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \tan (x - 1)}$

Step 5

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

$\frac{\sin (lim_{x \to \frac{π}{4}} x) - \cos (lim_{x \to \frac{π}{4}} x)}{\tan (lim_{x \to \frac{π}{4}} x - 1)}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{\sin (lim_{x \to \frac{π}{4}} x) - \cos (lim_{x \to \frac{π}{4}} x)}{\tan (lim_{x \to \frac{π}{4}} x - lim_{x \to \frac{π}{4}} 1)}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{4}$ .

$\frac{\sin (lim_{x \to \frac{π}{4}} x) - \cos (lim_{x \to \frac{π}{4}} x)}{\tan (lim_{x \to \frac{π}{4}} x - 1 \cdot 1)}$

Step 8

Evaluate the limits by plugging in $\frac{π}{4}$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{\sin (\frac{π}{4}) - \cos (lim_{x \to \frac{π}{4}} x)}{\tan (lim_{x \to \frac{π}{4}} x - 1 \cdot 1)}$

Step 8.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{\sin (\frac{π}{4}) - \cos (\frac{π}{4})}{\tan (lim_{x \to \frac{π}{4}} x - 1 \cdot 1)}$

Step 8.3

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{\sin (\frac{π}{4}) - \cos (\frac{π}{4})}{\tan (\frac{π}{4} - 1 \cdot 1)}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} - \cos (\frac{π}{4})}{\tan (\frac{π}{4} - 1 \cdot 1)}$

Step 9.1.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}}{\tan (\frac{π}{4} - 1 \cdot 1)}$

Step 9.1.3

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{2} - \sqrt{2}}{2}}{\tan (\frac{π}{4} - 1 \cdot 1)}$

Step 9.1.4

Rewrite $\frac{\sqrt{2} - \sqrt{2}}{2}$ in a factored form.

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Step 9.1.4.1

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$\frac{\frac{0}{2}}{\tan (\frac{π}{4} - 1 \cdot 1)}$

Step 9.1.4.2

Divide $0$ by $2$ .

$\frac{0}{\tan (\frac{π}{4} - 1 \cdot 1)}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 1$ by $1$ .

$\frac{0}{\tan (\frac{π}{4} - 1)}$

Step 9.2.2

Evaluate $\tan (\frac{π}{4} - 1)$ .

$\frac{0}{- 0.21795809}$

Step 9.3

Divide $0$ by $- 0.21795809$ .

$0$