What is the limit as x approaches infinity of (x^{3/2})/((x+1)^{3/2)} ?

The limit as x approaches infinity of (x^{3/2})/((x+1)^{3/2)} is 1

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limit as x approaches infinity of (x^{3/2})/((x+1)^{3/2)}

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

1

Solution steps

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

Simplify

\frac{x ^{\frac{3}{2}}}{( x + 1 ) ^{\frac{3}{2}}} : (\frac{x}{x + 1})^{\frac{3}{2}}

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

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

With the exception of indeterminate form

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

Divide by highest denominator power:

\frac{1}{1 + \frac{1}{x}}

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

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

x \to \infty lim (1) = 1

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

= (\frac{1}{1})^{\frac{3}{2}}

Simplify

(\frac{1}{1})^{\frac{3}{2}} : 1

= 1

What is the limit as x approaches infinity of (x^{3/2})/((x+1)^{3/2)} ?
The limit as x approaches infinity of (x^{3/2})/((x+1)^{3/2)} is 1