So this question is from one of my review practice problems for my midterm and there's no answer part for it and I was wondering how this question would be solved.
Lim x -> infinity 5x^3-2x+1 / x^4-x^3+5
Thanks!
edit: thank you!
So this question is from one of my review practice problems for my midterm and there's no answer part for it and I was wondering how this question would be solved.
Lim x -> infinity 5x^3-2x+1 / x^4-x^3+5
Thanks!
edit: thank you!
$\displaystyle \displaystyle \lim_{x\rightarrow \infty} \frac{5x^3 - 2x + 1}{x^4 - x^3 + 5}$
For questions like these (polynomials on the bottom), it helps to divide the numerator and denominator by term in the denominator with the highest power of x. In this case, $\displaystyle x^4 $
So we have
$\displaystyle \displaystyle \lim_{x\rightarrow \infty} \frac{\frac{5}{x} - \frac{2}{x^3} + \frac{1}{x^4}}{1 - \frac{1}{x} + \frac{5}{x^4}} $
You can apply the limit to each individual term in both the numerator and denominator.
Now any fraction of form $\displaystyle \frac{a}{x^n} \rightarrow 0$ as $\displaystyle x\rightarrow \infty $ for any real number a and positive integer n
You should get the resulting fraction $\displaystyle \frac{0-0+0}{1-0+0} = 0 $
Using exactly Gusbob's method in general you can get:
$\displaystyle \displaytype\lim_{x\to\infty}\frac{ax^n+ bx^{n-1}+ \cdot\cdot\cdot+ c}{ux^m+ vx^{m-1}+ \cdot\cdot\cdot+ w}$is equal to
i) infinity if n> m (numerator has higher degree)
ii) 0 if m< m (denominator has higher degree)
iii) $\displaystyle \frac{a}{u}$ if m= n. (numerator and denominator have same degree)