What is the limit as x approaches-3+of (-2x^2)/((x-1)(x+3)) ?

The limit as x approaches-3+of (-2x^2)/((x-1)(x+3)) is infinity

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limit as x approaches-3+of (-2x^2)/((x-1)(x+3))

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

\infty

Solution steps

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

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

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

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

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

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

Simplify

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

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

For

x

approaching

- 3

from the right

, x > - 3 \Rightarrow (- \frac{1}{x} + 1) (\frac{3}{x} + 1) < 0

The denominator is a negative quantity approaching 0 from the left

= - 2 (- \infty)

Apply Infinity Property:

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

= \infty

x \to 0 lim (2 x^{2} \cdot cot^{2} (x))

x \to \frac{π}{2} lim (sin (x) + 1)

x \to 0 lim (x^{2} ln (3 x))

x \to 1 lim (\frac{∣ x - 3 ∣}{x - 3})

x \to 0 lim (∣ sin (x) ∣ - \frac{1}{2})

What is the limit as x approaches-3+of (-2x^2)/((x-1)(x+3)) ?
The limit as x approaches-3+of (-2x^2)/((x-1)(x+3)) is infinity