Calculus Examples

$lim_{x \to 7} \sec (\frac{π x}{6})$

Step 1

Evaluate the limit.

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Step 1.1

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

$\sec (lim_{x \to 7} \frac{π x}{6})$

Step 1.2

Move the term $\frac{π}{6}$ outside of the limit because it is constant with respect to $x$ .

$\sec (\frac{π}{6} lim_{x \to 7} x)$

Step 2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\sec (\frac{π}{6} \cdot 7)$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{π}{6}$ and $7$ .

$\sec (\frac{π \cdot 7}{6})$

Step 3.2

Move $7$ to the left of $π$ .

$\sec (\frac{7 \cdot π}{6})$

Step 3.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the third quadrant.

$- \sec (\frac{π}{6})$

Step 3.4

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$- \frac{2}{\sqrt{3}}$

Step 3.5

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 3.6

Combine and simplify the denominator.

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Step 3.6.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.6.4

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

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.6.5

Add $1$ and $1$ .

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.6.6.2

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

$- \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.6.6.3

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

$- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.6.6.4

Cancel the common factor of $2$ .

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Step 3.6.6.4.1

Cancel the common factor.

$- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.6.6.4.2

Rewrite the expression.

$- \frac{2 \sqrt{3}}{3^{1}}$

Step 3.6.6.5

Evaluate the exponent.

$- \frac{2 \sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 \sqrt{3}}{3}$

Decimal Form:

$- 1.15470053 \dots$