Calculus Examples

$lim_{x \to 0} \cot (3 x) \sin (4 x) \cos (5 x)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} \cot (3 x) \cdot lim_{x \to 0} \sin (4 x) \cdot lim_{x \to 0} \cos (5 x)$

Step 2

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

$lim_{x \to 0} \cot (3 x) \cdot \sin (lim_{x \to 0} 4 x) \cdot lim_{x \to 0} \cos (5 x)$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$lim_{x \to 0} \cot (3 x) \cdot \sin (4 lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (5 x)$

Step 4

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

$lim_{x \to 0} \cot (3 x) \cdot \sin (4 lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} 5 x)$

Step 5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$lim_{x \to 0} \cot (3 x) \cdot \sin (4 lim_{x \to 0} x) \cdot \cos (5 lim_{x \to 0} x)$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 6.1

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

$lim_{x \to 0} \cot (3 x) \cdot \sin (4 \cdot 0) \cdot \cos (5 lim_{x \to 0} x)$

Step 6.2

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

$lim_{x \to 0} \cot (3 x) \cdot \sin (4 \cdot 0) \cdot \cos (5 \cdot 0)$

Step 7

Consider the left sided limit.

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

Step 8

As the $x$ values approach $0$ from the left, the function values decrease without bound.

$- \infty$

Step 9

Consider the right sided limit.

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

Step 10

As the $x$ values approach $0$ from the right, the function values increase without bound.

$\infty$

Step 11

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$