Calculus Examples

$lim_{x \to 0} 5 π (3 \cos (x) + 2 π \tan (\frac{5 π}{4 \sec (x)}) - 3)$

Step 1

Move the term $5 π$ outside of the limit because it is constant with respect to $x$ .

$5 π lim_{x \to 0} 3 \cos (x) + 2 π \tan (\frac{5 π}{4 \sec (x)}) - 3$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$5 π (lim_{x \to 0} 3 \cos (x) + lim_{x \to 0} 2 π \tan (\frac{5 π}{4 \sec (x)}) - lim_{x \to 0} 3)$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$5 π (3 lim_{x \to 0} \cos (x) + lim_{x \to 0} 2 π \tan (\frac{5 π}{4 \sec (x)}) - lim_{x \to 0} 3)$

Step 4

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

$5 π (3 \cos (lim_{x \to 0} x) + lim_{x \to 0} 2 π \tan (\frac{5 π}{4 \sec (x)}) - lim_{x \to 0} 3)$

Step 5

Move the term $2 π$ outside of the limit because it is constant with respect to $x$ .

$5 π (3 \cos (lim_{x \to 0} x) + 2 π lim_{x \to 0} \tan (\frac{5 π}{4 \sec (x)}) - lim_{x \to 0} 3)$

Step 6

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

$5 π (3 \cos (lim_{x \to 0} x) + 2 π \tan (lim_{x \to 0} \frac{5 π}{4 \sec (x)}) - lim_{x \to 0} 3)$

Step 7

Move the term $\frac{5 π}{4}$ outside of the limit because it is constant with respect to $x$ .

$5 π (3 \cos (lim_{x \to 0} x) + 2 π \tan (\frac{5 π}{4} lim_{x \to 0} \frac{1}{\sec (x)}) - lim_{x \to 0} 3)$

Step 8

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$5 π (3 \cos (lim_{x \to 0} x) + 2 π \tan (\frac{5 π}{4} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} \sec (x)}) - lim_{x \to 0} 3)$

Step 9

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$5 π (3 \cos (lim_{x \to 0} x) + 2 π \tan (\frac{5 π}{4} \cdot \frac{1}{lim_{x \to 0} \sec (x)}) - lim_{x \to 0} 3)$

Step 10

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

$5 π (3 \cos (lim_{x \to 0} x) + 2 π \tan (\frac{5 π}{4} \cdot \frac{1}{\sec (lim_{x \to 0} x)}) - lim_{x \to 0} 3)$

Step 11

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$5 π (3 \cos (lim_{x \to 0} x) + 2 π \tan (\frac{5 π}{4} \cdot \frac{1}{\sec (lim_{x \to 0} x)}) - 1 \cdot 3)$

Step 12

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 12.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$5 π (3 \cos (0) + 2 π \tan (\frac{5 π}{4} \cdot \frac{1}{\sec (lim_{x \to 0} x)}) - 1 \cdot 3)$

Step 12.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$5 π (3 \cos (0) + 2 π \tan (\frac{5 π}{4} \cdot \frac{1}{\sec (0)}) - 1 \cdot 3)$

Step 13

Simplify the answer.

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Step 13.1

Simplify each term.

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Step 13.1.1

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

$5 π (3 \cdot 1 + 2 π \tan (\frac{5 π}{4} \cdot \frac{1}{\sec (0)}) - 1 \cdot 3)$

Step 13.1.2

Multiply $3$ by $1$ .

$5 π (3 + 2 π \tan (\frac{5 π}{4} \cdot \frac{1}{\sec (0)}) - 1 \cdot 3)$

Step 13.1.3

Rewrite $\sec (0)$ in terms of sines and cosines.

$5 π (3 + 2 π \tan (\frac{5 π}{4} \cdot \frac{1}{\frac{1}{\cos (0)}}) - 1 \cdot 3)$

Step 13.1.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (0)}$ .

$5 π (3 + 2 π \tan (\frac{5 π}{4} (1 \cos (0))) - 1 \cdot 3)$

Step 13.1.5

Multiply $\cos (0)$ by $1$ .

$5 π (3 + 2 π \tan (\frac{5 π}{4} \cos (0)) - 1 \cdot 3)$

Step 13.1.6

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

$5 π (3 + 2 π \tan (\frac{5 π}{4} \cdot 1) - 1 \cdot 3)$

Step 13.1.7

Multiply $\frac{5 π}{4}$ by $1$ .

$5 π (3 + 2 π \tan (\frac{5 π}{4}) - 1 \cdot 3)$

Step 13.1.8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$5 π (3 + 2 π \tan (\frac{π}{4}) - 1 \cdot 3)$

Step 13.1.9

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$5 π (3 + 2 π \cdot 1 - 1 \cdot 3)$

Step 13.1.10

Multiply $2$ by $1$ .

$5 π (3 + 2 π - 1 \cdot 3)$

Step 13.1.11

Multiply $- 1$ by $3$ .

$5 π (3 + 2 π - 3)$

Step 13.2

Subtract $3$ from $3$ .

$5 π (2 π + 0)$

Step 13.3

Add $2 π$ and $0$ .

$5 π (2 π)$

Step 13.4

Multiply $5 π (2 π)$ .

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Step 13.4.1

Multiply $2$ by $5$ .

$10 π π$

Step 13.4.2

Raise $π$ to the power of $1$ .

$10 (π^{1} π)$

Step 13.4.3

Raise $π$ to the power of $1$ .

$10 (π^{1} π^{1})$

Step 13.4.4

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

$10 π^{1 + 1}$

Step 13.4.5

Add $1$ and $1$ .

$10 π^{2}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$10 π^{2}$

Decimal Form:

$98.69604401 \dots$