Calculus Examples

$lim_{x \to 0} \frac{1}{t \sec (t) \csc (4 t)}$

Step 1

Evaluate the limit of $\frac{1}{t \sec (t) \csc (4 t)}$ which is constant as $x$ approaches $0$ .

$\frac{1}{t \sec (t) \csc (4 t)}$

Step 2

Simplify the answer.

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Step 2.1

Separate fractions.

$\frac{1}{t \csc (4 t)} \cdot \frac{1}{\sec (t)}$

Step 2.2

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

$\frac{1}{t \csc (4 t)} \cdot \frac{1}{\frac{1}{\cos (t)}}$

Step 2.3

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

$\frac{1}{t \csc (4 t)} (1 \cos (t))$

Step 2.4

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

$\frac{1}{t \csc (4 t)} \cos (t)$

Step 2.5

Separate fractions.

$\frac{1}{t} \cdot \frac{1}{\csc (4 t)} \cos (t)$

Step 2.6

Rewrite $\csc (4 t)$ in terms of sines and cosines.

$\frac{1}{t} \cdot \frac{1}{\frac{1}{\sin (4 t)}} \cos (t)$

Step 2.7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (4 t)}$ .

$\frac{1}{t} (1 \sin (4 t)) \cos (t)$

Step 2.8

Multiply $\sin (4 t)$ by $1$ .

$\frac{1}{t} \sin (4 t) \cos (t)$

Step 2.9

Combine $\frac{1}{t}$ and $\sin (4 t)$ .

$\frac{\sin (4 t)}{t} \cos (t)$

Step 2.10

Combine $\frac{\sin (4 t)}{t}$ and $\cos (t)$ .

$\frac{\sin (4 t) \cos (t)}{t}$