Calculus Examples

$lim_{x \to \frac{3 π}{4}} \sec (x - \frac{π}{1})$

Step 1

Evaluate the limit.

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Step 1.1

Divide $π$ by $1$ .

$lim_{x \to \frac{3 π}{4}} \sec (x - π)$

Step 1.2

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

$\sec (lim_{x \to \frac{3 π}{4}} x - π)$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{3 π}{4}$ .

$\sec (lim_{x \to \frac{3 π}{4}} x - lim_{x \to \frac{3 π}{4}} π)$

Step 1.4

Evaluate the limit of $π$ which is constant as $x$ approaches $\frac{3 π}{4}$ .

$\sec (lim_{x \to \frac{3 π}{4}} x - π)$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{3 π}{4}$ for $x$ .

$\sec (\frac{3 π}{4} - π)$

Step 3

Simplify the answer.

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Step 3.1

To write $- π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sec (\frac{3 π}{4} - π \cdot \frac{4}{4})$

Step 3.2

Combine $- π$ and $\frac{4}{4}$ .

$\sec (\frac{3 π}{4} + \frac{- π \cdot 4}{4})$

Step 3.3

Combine the numerators over the common denominator.

$\sec (\frac{3 π - π \cdot 4}{4})$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $4$ by $- 1$ .

$\sec (\frac{3 π - 4 π}{4})$

Step 3.4.2

Subtract $4 π$ from $3 π$ .

$\sec (\frac{- π}{4})$

Step 3.5

Move the negative in front of the fraction.

$\sec (- \frac{π}{4})$

Step 3.6

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sec (\frac{7 π}{4})$

Step 3.7

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sec (\frac{π}{4})$

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Combine and simplify the denominator.

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Step 3.10.1

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

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

Step 3.10.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.10.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.10.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.10.5

Add $1$ and $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.10.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.10.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.10.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.10.6.4

Cancel the common factor of $2$ .

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Step 3.10.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.10.6.4.2

Rewrite the expression.

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

Step 3.10.6.5

Evaluate the exponent.

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

Step 3.11

Cancel the common factor of $2$ .

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Step 3.11.1

Cancel the common factor.

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

Step 3.11.2

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

$\sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$