What is the limit as n approaches infinity of (sqrt(n+2))/(sqrt((n+1)+2)) ?

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limit as n approaches infinity of (sqrt(n+2))/(sqrt((n+1)+2))

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

1

Solution steps

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

Simplify

\frac{n + 2}{n + 1 + 2} : \frac{n + 2}{n + 3}

= n \to \infty lim (\frac{n + 2}{n + 3})

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

With the exception of indeterminate form

= n \to \infty lim (\frac{n + 2}{n + 3})

Divide by highest denominator power:

\frac{1 + \frac{2}{n}}{1 + \frac{3}{n}}

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

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

n \to \infty lim (1 + \frac{2}{n}) = 1

n \to \infty lim (1 + \frac{3}{n}) = 1

= \frac{1}{1}

Simplify

= 1

x \to - 5 lim (x)

x \to 6 lim (\frac{2 - x - 2}{36 - x ^{2}})

x \to - 2 lim (2 x - 4)

x \to 0 lim (e^{x} x)

x \to 0 + lim (\frac{x ^{2} + 3 x + 4}{x})

What is the limit as n approaches infinity of (sqrt(n+2))/(sqrt((n+1)+2)) ?
The limit as n approaches infinity of (sqrt(n+2))/(sqrt((n+1)+2)) is 1