Find the limits. Write infty or -infty where appropriate. lim_{xrightarrow0^-}frac{x^2-3x+2}{x^3-4x}

Question

Find the limits. Write

\infty

or

- \infty

where appropriate.

lim_{x \to 0^{-}} \frac{x^{2} - 3 x + 2}{x^{3} - 4 x}

Answer 1

First let's write the limit
$lim_{x \to 0^{-}} \frac{x^{2} - 3 x + 2}{x^{3} - 4 x}$
$= lim_{h \to 0} \frac{x^{2} - 3 x + 2}{x^{3} - 4 x}$
$= lim_{h \to 0} \frac{- h^{2} - 3 (- h) + 2}{- h^{3} + 4 h}$
$= lim_{h \to 0} \frac{h^{2} + 3 g + 2}{- h^{3} + 4 h}$
Then, since h tends to 0, it must be fractional and less than 1
As $h < 1$
$h^{3}$
Hence, $h^{3} < 4 h$
$4 h - h^{3} > 0$
Hence, the denominator in the limit above approaches $0^{+}$
$L i m i t = \frac{0 + 0 + 2}{0^{+}} = \infty$
Result: $\infty$

Find the limits. Write infty or -infty where appropriate. lim_{xrightarrow0^-}frac{x^2-3x+2}{x^3-4x}

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