What is the limit as x approaches 2 of (sqrt(x+7)-\sqrt[3]{4x+19})/(x-2) ?

The limit as x approaches 2 of (sqrt(x+7)-\sqrt[3]{4x+19})/(x-2) is 1/54

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limit as x approaches 2 of (sqrt(x+7)-\sqrt[3]{4x+19})/(x-2)

\frac{1}{54}

Decimal Notation

0.01851 \dots

Solution steps

Apply L'Hopital's Rule

: x \to 2 lim \frac{\frac{1}{2 x + 7} - \frac{4}{3 ( 4 x + 19 ) ^{\frac{2}{3}}}}{1}

= x \to 2 lim \frac{\frac{1}{2 x + 7} - \frac{4}{3 ( 4 x + 19 ) ^{\frac{2}{3}}}}{1}

Plug in the value

x = 2

= \frac{\frac{1}{2 2 + 7} - \frac{4}{3 ( 4 \cdot 2 + 19 ) ^{\frac{2}{3}}}}{1}

Simplify

\frac{\frac{1}{2 2 + 7} - \frac{4}{3 ( 4 \cdot 2 + 19 ) ^{\frac{2}{3}}}}{1} : \frac{1}{54}

= \frac{1}{54}

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What is the limit as x approaches 2 of (sqrt(x+7)-\sqrt[3]{4x+19})/(x-2) ?
The limit as x approaches 2 of (sqrt(x+7)-\sqrt[3]{4x+19})/(x-2) is 1/54