What is the limit as x approaches 0 of (sqrt(1+x^2)-\sqrt[3]{1-2x})/x ?

The limit as x approaches 0 of (sqrt(1+x^2)-\sqrt[3]{1-2x})/x is 2/3

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limit as x approaches 0 of (sqrt(1+x^2)-\sqrt[3]{1-2x})/x

\frac{2}{3}

Decimal Notation

0.66666 \dots

Solution steps

Apply L'Hopital's Rule

: x \to 0 lim \frac{\frac{x}{1 + x ^{2}} + \frac{2}{3 ( 1 - 2 x ) ^{\frac{2}{3}}}}{1}

= x \to 0 lim \frac{\frac{x}{1 + x ^{2}} + \frac{2}{3 ( 1 - 2 x ) ^{\frac{2}{3}}}}{1}

Plug in the value

x = 0

= \frac{\frac{0}{1 + 0 ^{2}} + \frac{2}{3 ( 1 - 2 \cdot 0 ) ^{\frac{2}{3}}}}{1}

Simplify

\frac{\frac{0}{1 + 0 ^{2}} + \frac{2}{3 ( 1 - 2 \cdot 0 ) ^{\frac{2}{3}}}}{1} : \frac{2}{3}

= \frac{2}{3}

x \to 0 + lim (7 x ln (x))

x \to 0 + lim (e^{- \frac{1}{x}} \cdot x^{- 2})

x \to 0 lim (x^{2} sin (\frac{π}{x}))

x \to 3 lim (x - 7)

x \to 0 lim ((\frac{1 + 3 ^{x}}{2})^{\frac{1}{x}})

What is the limit as x approaches 0 of (sqrt(1+x^2)-\sqrt[3]{1-2x})/x ?
The limit as x approaches 0 of (sqrt(1+x^2)-\sqrt[3]{1-2x})/x is 2/3