What is the limit as x approaches infinity of (sqrt(x+a)+sqrt(x+b))/(sqrt(x)) ?

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limit as x approaches infinity of (sqrt(x+a)+sqrt(x+b))/(sqrt(x))

x \to \infty lim (\frac{x + a + x + b}{x})

2

Solution steps

x \to \infty lim (\frac{x + a + x + b}{x})

Expand

\frac{x + a + x + b}{x} : \frac{x + a}{x} + \frac{x + b}{x}

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

x \to a lim [f (x) \pm g (x)] = x \to a lim f (x) \pm x \to a lim g (x)

With the exception of indeterminate form

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

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

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

= 1 + 1

Simplify

= 2

What is the limit as x approaches infinity of (sqrt(x+a)+sqrt(x+b))/(sqrt(x)) ?
The limit as x approaches infinity of (sqrt(x+a)+sqrt(x+b))/(sqrt(x)) is 2