Calculus Examples

$lim_{x \to 2} \sec (\frac{π x}{3})$

Step 1

Evaluate the limit.

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Step 1.1

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

$\sec (lim_{x \to 2} \frac{π x}{3})$

Step 1.2

Move the term $\frac{π}{3}$ outside of the limit because it is constant with respect to $x$ .

$\sec (\frac{π}{3} lim_{x \to 2} x)$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\sec (\frac{π}{3} \cdot 2)$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{π}{3}$ and $2$ .

$\sec (\frac{π \cdot 2}{3})$

Step 3.2

Move $2$ to the left of $π$ .

$\sec (\frac{2 \cdot π}{3})$

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 second quadrant.

$- \sec (\frac{π}{3})$

Step 3.4

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

$- 1 \cdot 2$

Step 3.5

Multiply $- 1$ by $2$ .

$- 2$