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

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

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

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

\infty

Solution steps

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

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

Divide by highest denominator power:

\frac{x}{1 + \frac{3}{x}}

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

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

x \to \infty lim (x) = \infty

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

= 2 \cdot \frac{\infty}{1}

Simplify

2 \cdot \frac{\infty}{1} : \infty

= \infty

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