Calculus Examples

lim x \to 0 1 + {sin (4 x)}^{cot (x)}

$lim_{x \to 0} 1 + {\sin (4 x)}^{\cot (x)}$

Step 1

Set up the limit as a left-sided limit.

lim x \to 0^{-} 1 + {sin (4 x)}^{cot (x)}

$lim_{x \to 0^{-}} 1 + {\sin (4 x)}^{\cot (x)}$

Step 2

Evaluate the limits by plugging in the value for the variable.

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Step 2.1

Evaluate the limit of

1 + {sin (4 x)}^{cot (x)}

$1 + {\sin (4 x)}^{\cot (x)}$ by plugging in

0

$0$ for

x

$x$ .

1 + {sin (4 \cdot 0)}^{cot (0)}

$1 + {\sin (4 \cdot 0)}^{\cot (0)}$

Step 2.2

Rewrite

cot (0)

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

1 + {sin (4 \cdot 0)}^{\frac{cos (0)}{sin (0)}}

$1 + {\sin (4 \cdot 0)}^{\frac{\cos (0)}{\sin (0)}}$

Step 2.3

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

1 + {sin (4 \cdot 0)}^{\frac{cos (0)}{0}}

$1 + {\sin (4 \cdot 0)}^{\frac{\cos (0)}{0}}$

Step 2.4

Since

1 + {sin (4 \cdot 0)}^{\frac{cos (0)}{0}}

$1 + {\sin (4 \cdot 0)}^{\frac{\cos (0)}{0}}$ is undefined, the limit does not exist.

Does not exist

$Does not exist$

Does not exist

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

lim x \to 0^{+} 1 + {sin (4 x)}^{cot (x)}

$lim_{x \to 0^{+}} 1 + {\sin (4 x)}^{\cot (x)}$

Step 4

Evaluate the right-sided limit.

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Step 4.1

Evaluate the limit.

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Step 4.1.1

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

$0$ .

$lim_{x \to 0^{+}} 1 + lim_{x \to 0^{+}} {\sin (4 x)}^{\cot (x)}$

Step 4.1.2

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$0$ .

$1 + lim_{x \to 0^{+}} {\sin (4 x)}^{\cot (x)}$

Step 4.2

Use the properties of logarithms to simplify the limit.

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Step 4.2.1

Rewrite

${\sin (4 x)}^{\cot (x)}$ as

$e^{\ln ({\sin (4 x)}^{\cot (x)})}$ .

$1 + lim_{x \to 0^{+}} e^{\ln ({\sin (4 x)}^{\cot (x)})}$

Step 4.2.2

Expand

$\ln ({\sin (4 x)}^{\cot (x)})$ by moving

$\cot (x)$ outside the logarithm.

$1 + lim_{x \to 0^{+}} e^{\cot (x) \ln (\sin (4 x))}$

Step 4.3

Since the exponent

$\cot (x) \ln (\sin (4 x))$ approaches

$- \infty$ , the quantity

$e^{\cot (x) \ln (\sin (4 x))}$ approaches

$0$ .

$1 + 0$

Step 4.4

Add

$1$ and

$0$ .

$1$

Step 5

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$