What is the limit as x approaches-2-of (x^2)/(3x+6) ?

The limit as x approaches-2-of (x^2)/(3x+6) is -infinity

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limit as x approaches-2-of (x^2)/(3x+6)

x \to - 2 - lim (\frac{x ^{2}}{3 x + 6})

- \infty

Solution steps

x \to - 2 - lim (\frac{x ^{2}}{3 x + 6})

Factor out

3 x

from

3 x + 6 : 3 x (1 + \frac{6}{3 x})

= x \to - 2 - lim (\frac{x ^{2}}{3 x ( 1 + \frac{6}{3 x} )})

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

= \frac{1}{3} \cdot x \to - 2 - lim (\frac{x ^{2}}{x ( 1 + \frac{6}{3 x} )})

Simplify

\frac{x ^{2}}{x ( 1 + \frac{6}{3 x} )} : \frac{x ^{2}}{x + 2}

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

\frac{x ^{2}}{x + 2} = x^{2} \frac{1}{x + 2}

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

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

With the exception of indeterminate form

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

x \to - 2 - lim (x^{2}) = 4

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

= \frac{1}{3} \cdot 4 (- \infty)

Apply Infinity Property:

c \cdot (- \infty) = - \infty

= - \infty

t \to 1 lim (t^{2})

x \to 4 lim (5 x + 2)

x \to - 3 lim (2 x)

x \to - 1 + lim (x + 1)

x \to - \frac{π}{2} + lim (- 3 x - \frac{π}{2})

What is the limit as x approaches-2-of (x^2)/(3x+6) ?
The limit as x approaches-2-of (x^2)/(3x+6) is -infinity