Calculus Examples

lim x \to \frac{π}{3} - 5 tan (2 x) + cos (x)

$lim_{x \to \frac{π}{3}} - 5 \tan (2 x) + \cos (x)$

Step 1

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

\frac{π}{3}

$\frac{π}{3}$ .

- lim x \to \frac{π}{3} 5 tan (2 x) + lim x \to \frac{π}{3} cos (x)

$- lim_{x \to \frac{π}{3}} 5 \tan (2 x) + lim_{x \to \frac{π}{3}} \cos (x)$

Step 2

Move the term

5

$5$ outside of the limit because it is constant with respect to

x

$x$ .

- 5 lim x \to \frac{π}{3} tan (2 x) + lim x \to \frac{π}{3} cos (x)

$- 5 lim_{x \to \frac{π}{3}} \tan (2 x) + lim_{x \to \frac{π}{3}} \cos (x)$

Step 3

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

- 5 tan (lim x \to \frac{π}{3} 2 x) + lim x \to \frac{π}{3} cos (x)

$- 5 \tan (lim_{x \to \frac{π}{3}} 2 x) + lim_{x \to \frac{π}{3}} \cos (x)$

Step 4

Move the term

2

$2$ outside of the limit because it is constant with respect to

x

$x$ .

- 5 tan (2 lim x \to \frac{π}{3} x) + lim x \to \frac{π}{3} cos (x)

$- 5 \tan (2 lim_{x \to \frac{π}{3}} x) + lim_{x \to \frac{π}{3}} \cos (x)$

Step 5

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

$- 5 \tan (2 lim_{x \to \frac{π}{3}} x) + \cos (lim_{x \to \frac{π}{3}} x)$

Step 6

Evaluate the limits by plugging in

$\frac{π}{3}$ for all occurrences of

$x$ .

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Step 6.1

Evaluate the limit of

$x$ by plugging in

$\frac{π}{3}$ for

$x$ .

$- 5 \tan (2 \frac{π}{3}) + \cos (lim_{x \to \frac{π}{3}} x)$

Step 6.2

Evaluate the limit of

$x$ by plugging in

$\frac{π}{3}$ for

$x$ .

$- 5 \tan (2 \frac{π}{3}) + \cos (\frac{π}{3})$

Step 7

Simplify each term.

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Step 7.1

Combine

$2$ and

$\frac{π}{3}$ .

$- 5 \tan (\frac{2 π}{3}) + \cos (\frac{π}{3})$

Step 7.2

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

$- 5 (- \tan (\frac{π}{3})) + \cos (\frac{π}{3})$

Step 7.3

The exact value of

$\tan (\frac{π}{3})$ is

$\sqrt{3}$ .

$- 5 (- \sqrt{3}) + \cos (\frac{π}{3})$

Step 7.4

Multiply

$- 1$ by

$- 5$ .

$5 \sqrt{3} + \cos (\frac{π}{3})$

Step 7.5

The exact value of

$\cos (\frac{π}{3})$ is

$\frac{1}{2}$ .

$5 \sqrt{3} + \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{3} + \frac{1}{2}$

Decimal Form:

$9.16025403 \dots$