Calculus Examples

$lim_{x \to 3} \sec (\frac{7 π}{6})$

Step 1

Evaluate the limit of $\sec (\frac{7 π}{6})$ which is constant as $x$ approaches $3$ .

$\sec (\frac{7 π}{6})$

Step 2

Simplify the answer.

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Step 2.1

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 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.4.5

Add $1$ and $1$ .

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

Step 2.4.6

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

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Step 2.4.6.1

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 2.4.6.2

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

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

Step 2.4.6.3

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

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

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

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

Step 2.4.6.4.2

Rewrite the expression.

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

Step 2.4.6.5

Evaluate the exponent.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.15470053 \dots$