What is the limit as x approaches 0 of (5x-tan(5x))/(x^3) ?

The limit as x approaches 0 of (5x-tan(5x))/(x^3) is -125/3

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limit as x approaches 0 of (5x-tan(5x))/(x^3)

x \to 0 lim (\frac{5 x - tan ( 5 x )}{x ^{3}})

- \frac{125}{3}

Decimal Notation

- 41.66666 \dots

Solution steps

x \to 0 lim (\frac{5 x - tan ( 5 x )}{x ^{3}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- 5 tan ^{2} ( 5 x )}{3 x ^{2}})

= x \to 0 lim (\frac{- 5 tan ^{2} ( 5 x )}{3 x ^{2}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- 50 sec ^{2} ( 5 x ) tan ( 5 x )}{6 x})

= x \to 0 lim (\frac{- 50 sec ^{2} ( 5 x ) tan ( 5 x )}{6 x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- 50 ( 10 sec ^{2} ( 5 x ) tan ^{2} ( 5 x ) + 5 sec ^{4} ( 5 x ) )}{6})

= x \to 0 lim (\frac{- 50 ( 10 sec ^{2} ( 5 x ) tan ^{2} ( 5 x ) + 5 sec ^{4} ( 5 x ) )}{6})

Plug in the value

x = 0

= \frac{- 50 ( 10 sec ^{2} ( 5 \cdot 0 ) tan ^{2} ( 5 \cdot 0 ) + 5 sec ^{4} ( 5 \cdot 0 ) )}{6}

Simplify

\frac{- 50 ( 10 sec ^{2} ( 5 \cdot 0 ) tan ^{2} ( 5 \cdot 0 ) + 5 sec ^{4} ( 5 \cdot 0 ) )}{6} : - \frac{125}{3}

= - \frac{125}{3}

x \to - 7 lim (\frac{x ^{2} + 7 x}{x ^{2} - 2 x - 63})

x \to 0 lim (csc (x) sin (sin (x)))

t \to \infty lim (\frac{ln ( 3 t )}{t ^{2}})

x \to 0 lim (2 x + 3)

x \to 1 lim (x^{2} + 2 x + 5)

What is the limit as x approaches 0 of (5x-tan(5x))/(x^3) ?
The limit as x approaches 0 of (5x-tan(5x))/(x^3) is -125/3