What is the limit as x approaches 0 of (5x^n-3x^{n-1}+4x^{n-2})/(2x^n-6x^{n-2)} ?

The limit as x approaches 0 of (5x^n-3x^{n-1}+4x^{n-2})/(2x^n-6x^{n-2)} is -2/3

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limit as x approaches 0 of (5x^n-3x^{n-1}+4x^{n-2})/(2x^n-6x^{n-2)}

x \to 0 lim (\frac{5 x ^{n} - 3 x ^{n - 1} + 4 x ^{n - 2}}{2 x ^{n} - 6 x ^{n - 2}})

- \frac{2}{3}

Decimal

- 0.66666 \dots

Solution steps

x \to 0 lim (\frac{5 x ^{n} - 3 x ^{n - 1} + 4 x ^{n - 2}}{2 x ^{n} - 6 x ^{n - 2}})

Simplify

\frac{5 x ^{n} - 3 x ^{n - 1} + 4 x ^{n - 2}}{2 x ^{n} - 6 x ^{n - 2}} : \frac{5 x ^{2} - 3 x + 4}{2 ( - 3 + x ^{2} )}

= x \to 0 lim (\frac{5 x ^{2} - 3 x + 4}{2 ( - 3 + x ^{2} )})

Plug in the value

x = 0

= \frac{5 \cdot 0 ^{2} - 3 \cdot 0 + 4}{2 ( - 3 + 0 ^{2} )}

Simplify

\frac{5 \cdot 0 ^{2} - 3 \cdot 0 + 4}{2 ( - 3 + 0 ^{2} )} : - \frac{2}{3}

= - \frac{2}{3}

What is the limit as x approaches 0 of (5x^n-3x^{n-1}+4x^{n-2})/(2x^n-6x^{n-2)} ?
The limit as x approaches 0 of (5x^n-3x^{n-1}+4x^{n-2})/(2x^n-6x^{n-2)} is -2/3