What is the limit as x approaches 0 of x/(\sqrt[n]{1+ax)\sqrt[k]{1+bx-1}} ?

The limit as x approaches 0 of x/(\sqrt[n]{1+ax)\sqrt[k]{1+bx-1}} is 0

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limit as x approaches 0 of x/(\sqrt[n]{1+ax)\sqrt[k]{1+bx-1}}

x \to 0 lim (\frac{x}{n 1 + a x k 1 + b x - 1})

0

Solution steps

x \to 0 lim (\frac{x}{n 1 + a x k 1 + b x - 1})

Simplify

\frac{x}{n 1 + a x k 1 + b x - 1} : \frac{x ^{\frac{k - 1}{k}}}{b ^{\frac{1}{k}} ( 1 + a x ) ^{\frac{1}{n}}}

= x \to 0 lim (\frac{x ^{\frac{k - 1}{k}}}{b ^{\frac{1}{k}} \cdot ( 1 + a x ) ^{\frac{1}{n}}})

Plug in the value

x = 0

= \frac{0 ^{\frac{k - 1}{k}}}{b ^{\frac{1}{k}} \cdot ( 1 + a \cdot 0 ) ^{\frac{1}{n}}}

Simplify

\frac{0 ^{\frac{k - 1}{k}}}{b ^{\frac{1}{k}} \cdot ( 1 + a \cdot 0 ) ^{\frac{1}{n}}} : 0

= 0

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What is the limit as x approaches 0 of x/(\sqrt[n]{1+ax)\sqrt[k]{1+bx-1}} ?
The limit as x approaches 0 of x/(\sqrt[n]{1+ax)\sqrt[k]{1+bx-1}} is 0