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

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limit as x approaches infinity of (sqrt(1+\sqrt{1/x)}-1)sqrt(x)

x \to \infty lim 1 + \frac{1}{x} - 1 x

\frac{1}{2}

Decimal

0.5

Solution steps

x \to \infty lim 1 + \frac{1}{x} - 1 x

Multiply by the conjugate of

1 + \frac{1}{x} - 1 : \frac{\frac{1}{x}}{1 + \frac{1}{x} + 1}

= x \to \infty lim \frac{\frac{1}{x}}{1 + \frac{1}{x} + 1} x

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= x \to \infty lim \frac{\frac{1}{x} x}{1 + \frac{1}{x} + 1}

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 )}{lim _{x \to \infty} ( 1 + \frac{1}{x} + 1 )}

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

x \to \infty lim 1 + \frac{1}{x} + 1 = 2

= \frac{1}{2}

y^{^{'}} = e^{x - y}

\int_{- 2}^{2} 5 - 4 - x^{2} d x

\int \frac{- 1}{x + 1} d x

f (x) = x^{2} - 2 x - 3

\frac{\partial}{\partial x} (x^{2} e^{x + y} + 2 y cos (x y) - x y^{2} sin (x y))

What is the limit as x approaches infinity of (sqrt(1+\sqrt{1/x)}-1)sqrt(x) ?
The limit as x approaches infinity of (sqrt(1+\sqrt{1/x)}-1)sqrt(x) is 1/2