Calculus Examples

$lim_{x \to \frac{π}{4}} \frac{1 + \tan (x)}{\csc (x) + 2}$

Step 1

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

$\frac{lim_{x \to \frac{π}{4}} 1 + \tan (x)}{lim_{x \to \frac{π}{4}} \csc (x) + 2}$

Step 2

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

$\frac{lim_{x \to \frac{π}{4}} 1 + lim_{x \to \frac{π}{4}} \tan (x)}{lim_{x \to \frac{π}{4}} \csc (x) + 2}$

Step 3

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

$\frac{1 + lim_{x \to \frac{π}{4}} \tan (x)}{lim_{x \to \frac{π}{4}} \csc (x) + 2}$

Step 4

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

$\frac{1 + \tan (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \csc (x) + 2}$

Step 5

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

$\frac{1 + \tan (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \csc (x) + lim_{x \to \frac{π}{4}} 2}$

Step 6

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

$\frac{1 + \tan (lim_{x \to \frac{π}{4}} x)}{\csc (lim_{x \to \frac{π}{4}} x) + lim_{x \to \frac{π}{4}} 2}$

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $\frac{π}{4}$ .

$\frac{1 + \tan (lim_{x \to \frac{π}{4}} x)}{\csc (lim_{x \to \frac{π}{4}} x) + 2}$

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 $x$ .

$\frac{1 + \tan (\frac{π}{4})}{\csc (lim_{x \to \frac{π}{4}} x) + 2}$

Step 8.2

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

$\frac{1 + \tan (\frac{π}{4})}{\csc (\frac{π}{4}) + 2}$

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 $1$ .

$\frac{1 + 1}{\csc (\frac{π}{4}) + 2}$

Step 9.1.2

Add $1$ and $1$ .

$\frac{2}{\csc (\frac{π}{4}) + 2}$

Step 9.2

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

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

Step 9.3

Multiply $\frac{2}{\sqrt{2} + 2}$ by $\frac{\sqrt{2} - 2}{\sqrt{2} - 2}$ .

$\frac{2}{\sqrt{2} + 2} \cdot \frac{\sqrt{2} - 2}{\sqrt{2} - 2}$

Step 9.4

Multiply $\frac{2}{\sqrt{2} + 2}$ by $\frac{\sqrt{2} - 2}{\sqrt{2} - 2}$ .

$\frac{2 (\sqrt{2} - 2)}{(\sqrt{2} + 2) (\sqrt{2} - 2)}$

Step 9.5

Expand the denominator using the FOIL method.

$\frac{2 (\sqrt{2} - 2)}{{\sqrt{2}}^{2} + \sqrt{2} \cdot - 2 + 2 \sqrt{2} - 4}$

Step 9.6

Simplify.

$\frac{2 (\sqrt{2} - 2)}{- 2}$

Step 9.7

Move the negative one from the denominator of $\frac{\sqrt{2} - 2}{- 1}$ .

$- 1 \cdot (\sqrt{2} - 2)$

Step 9.8

Rewrite $- 1 \cdot (\sqrt{2} - 2)$ as $- (\sqrt{2} - 2)$ .

$- (\sqrt{2} - 2)$

Step 9.9

Apply the distributive property.

$- \sqrt{2} - - 2$

Step 9.10

Multiply $- 1$ by $- 2$ .

$- \sqrt{2} + 2$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2} + 2$

Decimal Form:

$0.58578643 \dots$