Calculus Examples

lim_{x \to 0} \frac{\cot (4 x)}{\cot (3 x)}

$lim_{x \to 0} \frac{\cot (4 x)}{\cot (3 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite

\cot (3 x)

$\cot (3 x)$ in terms of sines and cosines.

lim_{x \to 0} \frac{\cot (4 x)}{\frac{\cos (3 x)}{\sin (3 x)}}

$lim_{x \to 0} \frac{\cot (4 x)}{\frac{\cos (3 x)}{\sin (3 x)}}$

Step 1.2

Rewrite

\cot (4 x)

$\cot (4 x)$ in terms of sines and cosines.

lim_{x \to 0} \frac{\frac{\cos (4 x)}{\sin (4 x)}}{\frac{\cos (3 x)}{\sin (3 x)}}

$lim_{x \to 0} \frac{\frac{\cos (4 x)}{\sin (4 x)}}{\frac{\cos (3 x)}{\sin (3 x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by

\frac{\cos (3 x)}{\sin (3 x)}

$\frac{\cos (3 x)}{\sin (3 x)}$ .

lim_{x \to 0} \frac{\cos (4 x)}{\sin (4 x)} \cdot \frac{\sin (3 x)}{\cos (3 x)}

$lim_{x \to 0} \frac{\cos (4 x)}{\sin (4 x)} \cdot \frac{\sin (3 x)}{\cos (3 x)}$

Step 1.4

Simplify.

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Step 1.4.1

Convert from

\frac{\cos (4 x)}{\sin (4 x)}

$\frac{\cos (4 x)}{\sin (4 x)}$ to

\cot (4 x)

$\cot (4 x)$ .

lim_{x \to 0} \cot (4 x) \frac{\sin (3 x)}{\cos (3 x)}

$lim_{x \to 0} \cot (4 x) \frac{\sin (3 x)}{\cos (3 x)}$

Step 1.4.2

Convert from

\frac{\sin (3 x)}{\cos (3 x)}

$\frac{\sin (3 x)}{\cos (3 x)}$ to

\tan (3 x)

$\tan (3 x)$ .

lim_{x \to 0} \cot (4 x) \tan (3 x)

$lim_{x \to 0} \cot (4 x) \tan (3 x)$

lim_{x \to 0} \cot (4 x) \tan (3 x)

$lim_{x \to 0} \cot (4 x) \tan (3 x)$

lim_{x \to 0} \cot (4 x) \tan (3 x)

$lim_{x \to 0} \cot (4 x) \tan (3 x)$

Step 2

Consider the left sided limit.

lim_{x \to 0^{-}} \cot (4 x) \tan (3 x)

$lim_{x \to 0^{-}} \cot (4 x) \tan (3 x)$

Step 3

Make a table to show the behavior of the function

\cot (4 x) \tan (3 x)

$\cot (4 x) \tan (3 x)$ as

x

$x$ approaches

0

$0$ from the left.

\begin{array}{c} x & \cot (4 x) \tan (3 x) \\ - 0.1 & 0.73164903 \\ - 0.01 & 0.74982491 \\ - 0.001 & 0.74999824 \end{array}

$\begin{array}{c} x & \cot (4 x) \tan (3 x) \\ - 0.1 & 0.73164903 \\ - 0.01 & 0.74982491 \\ - 0.001 & 0.74999824 \end{array}$

Step 4

As the

x

$x$ values approach

0

$0$ , the function values approach

0.75

$0.75$ . Thus, the limit of

\cot (4 x) \tan (3 x)

$\cot (4 x) \tan (3 x)$ as

x

$x$ approaches

0

$0$ from the left is

0.75

$0.75$ .

0.75

$0.75$

Step 5

Consider the right sided limit.

lim_{x \to 0^{+}} \cot (4 x) \tan (3 x)

$lim_{x \to 0^{+}} \cot (4 x) \tan (3 x)$

Step 6

Make a table to show the behavior of the function

\cot (4 x) \tan (3 x)

$\cot (4 x) \tan (3 x)$ as

x

$x$ approaches

0

$0$ from the right.

\begin{array}{c} x & \cot (4 x) \tan (3 x) \\ 0.1 & 0.73164903 \\ 0.01 & 0.74982491 \\ 0.001 & 0.74999824 \end{array}

$\begin{array}{c} x & \cot (4 x) \tan (3 x) \\ 0.1 & 0.73164903 \\ 0.01 & 0.74982491 \\ 0.001 & 0.74999824 \end{array}$

Step 7

As the

x

$x$ values approach

0

$0$ , the function values approach

0.75

$0.75$ . Thus, the limit of

\cot (4 x) \tan (3 x)

$\cot (4 x) \tan (3 x)$ as

x

$x$ approaches

0

$0$ from the right is

0.75

$0.75$ .

0.75

$0.75$