What is the limit as x approaches 0 of x/(x+sin(2x)) ?

The limit as x approaches 0 of x/(x+sin(2x)) is 1/3

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limit as x approaches 0 of x/(x+sin(2x))

x \to 0 lim (\frac{x}{x + sin ( 2 x )})

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

x \to 0 lim (\frac{x}{x + sin ( 2 x )})

Divide by highest denominator power:

\frac{1}{1 + \frac{s i n ( 2 x )}{x}}

= x \to 0 lim (\frac{1}{1 + \frac{s i n ( 2 x )}{x}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to 0} ( 1 )}{lim _{x \to 0} ( 1 + \frac{s i n ( 2 x )}{x} )}

x \to 0 lim (1) = 1

x \to 0 lim (1 + \frac{sin ( 2 x )}{x}) = 3

= \frac{1}{3}

x \to 1 lim ((x - 1)^{x})

x \to 0 lim (\frac{x}{5 + sin ( x )})

x \to 2 - lim (\frac{10}{x ^{2} - 4})

x \to 0 lim (x cot (π x))

x \to 4 lim (3 x^{2} - 2 x + 6)

What is the limit as x approaches 0 of x/(x+sin(2x)) ?
The limit as x approaches 0 of x/(x+sin(2x)) is 1/3