What is the limit as x approaches infinity of (1-x\sqrt[x]{x})/(x-sin(x)) ?

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limit as x approaches infinity of (1-x\sqrt[x]{x})/(x-sin(x))

- 1

Solution steps

Divide by highest denominator power:

\frac{\frac{1}{x} - x ^{\frac{1}{x}}}{1 - \frac{s i n ( x )}{x}}

= x \to \infty lim (\frac{\frac{1}{x} - x ^{\frac{1}{x}}}{1 - \frac{s i n ( x )}{x}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to \infty} ( \frac{1}{x} - x ^{\frac{1}{x}} )}{lim _{x \to \infty} ( 1 - \frac{s i n ( x )}{x} )}

x \to \infty lim (\frac{1}{x} - x^{\frac{1}{x}}) = - 1

x \to \infty lim (1 - \frac{sin ( x )}{x}) = 1

= \frac{- 1}{1}

Simplify

= - 1

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What is the limit as x approaches infinity of (1-x\sqrt[x]{x})/(x-sin(x)) ?
The limit as x approaches infinity of (1-x\sqrt[x]{x})/(x-sin(x)) is -1