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

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

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

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

1

Solution steps

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

Simplify

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

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

Divide by highest denominator power:

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

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

x \to \infty lim (1) = 1

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

= \frac{1}{1}

Simplify

= 1

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