Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

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 $x$ .

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

Step 8.2

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

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

Step 8.3

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

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

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 $\tan (\frac{π}{4})$ is $1$ .

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

Step 9.1.2

Multiply $- 1$ by $1$ .

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

Step 9.1.3

Subtract $1$ from $1$ .

$\frac{0}{\sin (\frac{π}{4}) + \cos (\frac{π}{4})}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$\frac{0}{\frac{\sqrt{2}}{2} + \cos (\frac{π}{4})}$

Step 9.2.2

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

$\frac{0}{\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}}$

Step 9.2.3

Combine the numerators over the common denominator.

$\frac{0}{\frac{\sqrt{2} + \sqrt{2}}{2}}$

Step 9.2.4

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

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Step 9.2.4.1

Add $\sqrt{2}$ and $\sqrt{2}$ .

$\frac{0}{\frac{2 \sqrt{2}}{2}}$

Step 9.2.4.2

Reduce the expression by cancelling the common factors.

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Step 9.2.4.2.1

Reduce the expression $\frac{2 \sqrt{2}}{2}$ by cancelling the common factors.

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Step 9.2.4.2.1.1

Cancel the common factor.

$\frac{0}{\frac{2 \sqrt{2}}{2}}$

Step 9.2.4.2.1.2

Rewrite the expression.

$\frac{0}{\frac{\sqrt{2}}{1}}$

Step 9.2.4.2.2

Divide $\sqrt{2}$ by $1$ .

$\frac{0}{\sqrt{2}}$

Step 9.3

Divide $0$ by $\sqrt{2}$ .

$0$