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

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

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

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

3

Solution steps

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

Factor out

x

from

x + sin (2 x) : x (1 + \frac{sin ( 2 x )}{x})

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

Cancel the common factor:

x

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

x \to a lim [f (x) \pm g (x)] = x \to a lim f (x) \pm x \to a lim g (x)

With the exception of indeterminate form

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

x \to 0 lim (1) = 1

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

= 1 + 2

Simplify

= 3

x \to 0 lim (\frac{2}{x} - \frac{1}{e ^{x} - 1})

x \to 0 + lim ((x)^{s i n (x)})

x \to 3 lim (45)

x \to - 1 lim (1 - 3 x^{2} + 2 x^{3})

x \to 1 lim (\frac{1}{1 - x} ln (x))

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