Calculus Examples

$lim_{x \to 5} \cos (\frac{7 π}{x} - \frac{2 π}{5})$

Step 1

Evaluate the limit.

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Step 1.1

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

$\cos (lim_{x \to 5} \frac{7 π}{x} - \frac{2 π}{5})$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\cos (lim_{x \to 5} \frac{7 π}{x} - lim_{x \to 5} \frac{2 π}{5})$

Step 1.3

Move the term $7 π$ outside of the limit because it is constant with respect to $x$ .

$\cos (7 π lim_{x \to 5} \frac{1}{x} - lim_{x \to 5} \frac{2 π}{5})$

Step 1.4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\cos (7 π \frac{lim_{x \to 5} 1}{lim_{x \to 5} x} - lim_{x \to 5} \frac{2 π}{5})$

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$\cos (7 π \frac{1}{lim_{x \to 5} x} - lim_{x \to 5} \frac{2 π}{5})$

Step 1.6

Evaluate the limit of $\frac{2 π}{5}$ which is constant as $x$ approaches $5$ .

$\cos (7 π \frac{1}{lim_{x \to 5} x} - \frac{2 π}{5})$

Step 2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\cos (7 π \frac{1}{5} - \frac{2 π}{5})$

Step 3

Simplify the answer.

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Step 3.1

Multiply $7 π \frac{1}{5}$ .

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Step 3.1.1

Combine $\frac{1}{5}$ and $7$ .

$\cos (\frac{7}{5} π - \frac{2 π}{5})$

Step 3.1.2

Combine $\frac{7}{5}$ and $π$ .

$\cos (\frac{7 π}{5} - \frac{2 π}{5})$

Step 3.2

Combine the numerators over the common denominator.

$\cos (\frac{7 π - 2 π}{5})$

Step 3.3

Subtract $2 π$ from $7 π$ .

$\cos (\frac{5 π}{5})$

Step 3.4

Cancel the common factor of $5$ .

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Step 3.4.1

Cancel the common factor.

$\cos (\frac{5 π}{5})$

Step 3.4.2

Divide $π$ by $1$ .

$\cos (π)$

Step 3.5

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

$- \cos (0)$

Step 3.6

The exact value of $\cos (0)$ is $1$ .

$- 1 \cdot 1$

Step 3.7

Multiply $- 1$ by $1$ .

$- 1$