Calculus Examples

$lim_{t \to 0} \frac{\tan (6 t)}{\sin (6 t)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (6 t)$ in terms of sines and cosines.

$lim_{t \to 0} \frac{\frac{\sin (6 t)}{\cos (6 t)}}{\sin (6 t)}$

Step 1.2

Rewrite $\frac{\frac{\sin (6 t)}{\cos (6 t)}}{\sin (6 t)}$ as a product.

$lim_{t \to 0} \frac{\sin (6 t)}{\cos (6 t)} \cdot \frac{1}{\sin (6 t)}$

Step 1.3

Cancel the common factor of $\sin (6 t)$ .

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Step 1.3.1

Cancel the common factor.

$lim_{t \to 0} \frac{\sin (6 t)}{\cos (6 t)} \cdot \frac{1}{\sin (6 t)}$

Step 1.3.2

Rewrite the expression.

$lim_{t \to 0} \frac{1}{\cos (6 t)}$

Step 1.4

Convert from $\frac{1}{\cos (6 t)}$ to $\sec (6 t)$ .

$lim_{t \to 0} \sec (6 t)$

Step 2

Evaluate the limit.

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Step 2.1

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

$\sec (lim_{t \to 0} 6 t)$

Step 2.2

Move the term $6$ outside of the limit because it is constant with respect to $t$ .

$\sec (6 lim_{t \to 0} t)$

Step 3

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

$\sec (6 \cdot 0)$

Step 4

Simplify the answer.

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Step 4.1

Multiply $6$ by $0$ .

$\sec (0)$

Step 4.2

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

$1$