What is the limit as x approaches 1 of (sqrt(x)+sqrt(4x+5)-sqrt(3x+13))/(x-1) ?

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limit as x approaches 1 of (sqrt(x)+sqrt(4x+5)-sqrt(3x+13))/(x-1)

x \to 1 lim (\frac{x + 4 x + 5 - 3 x + 13}{x - 1})

\frac{19}{24}

Decimal

0.79166 \dots

Solution steps

x \to 1 lim (\frac{x + 4 x + 5 - 3 x + 13}{x - 1})

Apply L'Hopital's Rule

= x \to 1 lim (\frac{\frac{1}{2 x} + \frac{2}{4 x + 5} - \frac{3}{2 3 x + 13}}{1})

Plug in the value

x = 1

= \frac{\frac{1}{2 1} + \frac{2}{4 \cdot 1 + 5} - \frac{3}{2 3 \cdot 1 + 13}}{1}

Simplify

\frac{\frac{1}{2 1} + \frac{2}{4 \cdot 1 + 5} - \frac{3}{2 3 \cdot 1 + 13}}{1} : \frac{19}{24}

= \frac{19}{24}

What is the limit as x approaches 1 of (sqrt(x)+sqrt(4x+5)-sqrt(3x+13))/(x-1) ?
The limit as x approaches 1 of (sqrt(x)+sqrt(4x+5)-sqrt(3x+13))/(x-1) is 19/24