Calculus Examples

$lim_{t \to 0} \cos (\frac{π}{\sqrt{19 - 3 \sec (2 t)}})$

Step 1

Evaluate the limit.

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Step 1.1

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

$\cos (lim_{t \to 0} \frac{π}{\sqrt{19 - 3 \sec (2 t)}})$

Step 1.2

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

$\cos (π lim_{t \to 0} \frac{1}{\sqrt{19 - 3 \sec (2 t)}})$

Step 1.3

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

$\cos (π \frac{lim_{t \to 0} 1}{lim_{t \to 0} \sqrt{19 - 3 \sec (2 t)}})$

Step 1.4

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

$\cos (π \frac{1}{lim_{t \to 0} \sqrt{19 - 3 \sec (2 t)}})$

Step 1.5

Move the limit under the radical sign.

$\cos (π \frac{1}{\sqrt{lim_{t \to 0} 19 - 3 \sec (2 t)}})$

Step 1.6

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

$\cos (π \frac{1}{\sqrt{lim_{t \to 0} 19 - lim_{t \to 0} 3 \sec (2 t)}})$

Step 1.7

Evaluate the limit of $19$ which is constant as $t$ approaches $0$ .

$\cos (π \frac{1}{\sqrt{19 - lim_{t \to 0} 3 \sec (2 t)}})$

Step 1.8

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

$\cos (π \frac{1}{\sqrt{19 - (3 lim_{t \to 0} \sec (2 t))}})$

Step 1.9

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

$\cos (π \frac{1}{\sqrt{19 - (3 \sec (lim_{t \to 0} 2 t))}})$

Step 1.10

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

$\cos (π \frac{1}{\sqrt{19 - (3 \sec (2 lim_{t \to 0} t))}})$

Step 2

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

$\cos (π \frac{1}{\sqrt{19 - (3 \sec (2 \cdot 0))}})$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Multiply $- 1$ by $3$ .

$\cos (π \frac{1}{\sqrt{19 - 3 \sec (2 \cdot 0)}})$

Step 3.1.2

Multiply $2$ by $0$ .

$\cos (π \frac{1}{\sqrt{19 - 3 \sec (0)}})$

Step 3.1.3

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

$\cos (π \frac{1}{\sqrt{19 - 3 \cdot 1}})$

Step 3.1.4

Multiply $- 3$ by $1$ .

$\cos (π \frac{1}{\sqrt{19 - 3}})$

Step 3.1.5

Subtract $3$ from $19$ .

$\cos (π \frac{1}{\sqrt{16}})$

Step 3.1.6

Rewrite $16$ as $4^{2}$ .

$\cos (π \frac{1}{\sqrt{4^{2}}})$

Step 3.1.7

Pull terms out from under the radical, assuming positive real numbers.

$\cos (π \frac{1}{4})$

Step 3.2

Combine $π$ and $\frac{1}{4}$ .

$\cos (\frac{π}{4})$

Step 3.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$