Calculus Examples

$lim_{x \to 0} \frac{5 \sin (x)}{3} + \frac{2}{1 - \cos (x)}$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{x \to 0} \frac{5 \sin (x)}{3} + lim_{x \to 0} \frac{2}{1 - \cos (x)}$

Step 1.2

Move the term $\frac{5}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{3} lim_{x \to 0} \sin (x) + lim_{x \to 0} \frac{2}{1 - \cos (x)}$

Step 1.3

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

$\frac{5}{3} \sin (lim_{x \to 0} x) + lim_{x \to 0} \frac{2}{1 - \cos (x)}$

Step 2

Since the numerator is positive and the denominator $1 - \cos (x)$ approaches zero and is greater than zero for $x$ near $0$ on both sides, the function increases without bound.

$\frac{5}{3} \sin (lim_{x \to 0} x) + \infty$

Step 3

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

$\frac{5}{3} \sin (0) + \infty$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

The exact value of $\sin (0)$ is $0$ .

$\frac{5}{3} \cdot 0 + \infty$

Step 4.1.2

Multiply $\frac{5}{3}$ by $0$ .

$0 + \infty$

Step 4.2

Add $0$ and $\infty$ .

$\infty$