What is the limit as x approaches 0 of (4x-2x^2)/(sin(x)) ?

The limit as x approaches 0 of (4x-2x^2)/(sin(x)) is 4

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limit as x approaches 0 of (4x-2x^2)/(sin(x))

x \to 0 lim (\frac{4 x - 2 x ^{2}}{sin ( x )})

4

Solution steps

x \to 0 lim (\frac{4 x - 2 x ^{2}}{sin ( x )})

Expand

\frac{4 x - 2 x ^{2}}{sin ( x )} : \frac{4 x}{sin ( x )} - \frac{2 x ^{2}}{sin ( x )}

= x \to 0 lim (\frac{4 x}{sin ( x )} - \frac{2 x ^{2}}{sin ( 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 (\frac{4 x}{sin ( x )}) - x \to 0 lim (\frac{2 x ^{2}}{sin ( x )})

x \to 0 lim (\frac{4 x}{sin ( x )}) = 4

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

= 4 - 0

Simplify

= 4

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

x \to + 0 + lim (sin (x) n x)

n \to 1 lim (\frac{n ^{3} - 1}{n - 1})

x \to 0 lim (\frac{x ^{2} - 2 x}{3 x})

x \to 0 lim (\frac{x - 2}{∣ x ∣ - 2})

What is the limit as x approaches 0 of (4x-2x^2)/(sin(x)) ?
The limit as x approaches 0 of (4x-2x^2)/(sin(x)) is 4