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

The limit as x approaches 3 of (1/3 sqrt(x)-sqrt(1/3))/(x-3) is 1/(6sqrt(3))

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

x \to 3 lim \frac{\frac{1}{3} x - \frac{1}{3}}{x - 3}

\frac{1}{6 3}

Decimal

0.09622 \dots

Solution steps

x \to 3 lim \frac{\frac{1}{3} x - \frac{1}{3}}{x - 3}

Multiply by the conjugate of

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

= x \to 3 lim \frac{\frac{\frac{1}{9} x - \frac{1}{3}}{\frac{1}{3} x + \frac{1}{3}}}{x - 3}

Simplify

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

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

Plug in the value

x = 3

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

Simplify

= \frac{1}{6 3}

What is the limit as x approaches 3 of (1/3 sqrt(x)-sqrt(1/3))/(x-3) ?
The limit as x approaches 3 of (1/3 sqrt(x)-sqrt(1/3))/(x-3) is 1/(6sqrt(3))