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

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

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limit as x approaches 0 of ((x-tan(3x)))/((x+tan(2x)))

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

- \frac{2}{3}

Decimal

- 0.66666 \dots

Solution steps

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

Divide by highest denominator power:

\frac{1 - \frac{t a n ( 3 x )}{x}}{1 + \frac{t a n ( 2 x )}{x}}

= x \to 0 lim (\frac{1 - \frac{t a n ( 3 x )}{x}}{1 + \frac{t a 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 - \frac{t a n ( 3 x )}{x} )}{lim _{x \to 0} ( 1 + \frac{t a n ( 2 x )}{x} )}

x \to 0 lim (1 - \frac{tan ( 3 x )}{x}) = - 2

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

= \frac{- 2}{3}

Simplify

= - \frac{2}{3}

x \to - 5 lim (\frac{25 - x ^{2}}{5 + x})

x \to 4 lim (\frac{( x - 4 )}{x + 5})

x \to 2 lim (\frac{x ( x + 3 ) + 1}{x ^{2} - 4 x + 4})

x \to 1 lim (\frac{120 x ^{2}}{x ^{2} + 4})

x \to 0 + lim (\frac{6 csc ( x )}{cot ( x )})

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