What is the limit as x approaches 0 of (1/x)(1/3-1/(sqrt(x+9))) ?

The limit as x approaches 0 of (1/x)(1/3-1/(sqrt(x+9))) is 1/54

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limit as x approaches 0 of (1/x)(1/3-1/(sqrt(x+9)))

x \to 0 lim ((\frac{1}{x}) (\frac{1}{3} - \frac{1}{x + 9}))

\frac{1}{54}

Decimal

0.01851 \dots

Solution steps

x \to 0 lim (\frac{1}{x} (\frac{1}{3} - \frac{1}{x + 9}))

Rationalize numerator

\frac{1}{x} (\frac{1}{3} - \frac{1}{x + 9}) : \frac{1}{3 ( x + 9 + 3 ) x + 9}

= x \to 0 lim (\frac{1}{3 ( x + 9 + 3 ) x + 9})

Plug in the value

x = 0

= \frac{1}{3 ( 0 + 9 + 3 ) 0 + 9}

Simplify

\frac{1}{3 ( 0 + 9 + 3 ) 0 + 9} : \frac{1}{54}

= \frac{1}{54}

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

x \to - \frac{π}{2} - lim (sec (x))

x \to 2 - lim (x - 3)

x \to \frac{π}{3} - lim (sin (3 x))

x \to - 1 lim (2 x - 2)

What is the limit as x approaches 0 of (1/x)(1/3-1/(sqrt(x+9))) ?
The limit as x approaches 0 of (1/x)(1/3-1/(sqrt(x+9))) is 1/54