1. Limits Question

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!

2. $\displaystyle \lim_{x\rightarrow \infty} \frac{5x^3 - 2x + 1}{x^4 - x^3 + 5}$

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, $x^4$

So we have

$\displaystyle \lim_{x\rightarrow \infty} \frac{\frac{5}{x} - \frac{2}{x^3} + \frac{1}{x^4}}{1 - \frac{1}{x} + \frac{5}{x^4}}$

You can apply the limit to each individual term in both the numerator and denominator.

Now any fraction of form $\frac{a}{x^n} \rightarrow 0$ as $x\rightarrow \infty$ for any real number a and positive integer n

You should get the resulting fraction $\frac{0-0+0}{1-0+0} = 0$

3. Using exactly Gusbob's method in general you can get:

$\displaytype\lim_{x\to\infty}\frac{ax^n+ bx^{n-1}+ \cdot\cdot\cdot+ c}{ux^m+ vx^{m-1}+ \cdot\cdot\cdot+ w}$is equal to

i) infinity if n> m (numerator has higher degree)

ii) 0 if m< m (denominator has higher degree)

iii) $\frac{a}{u}$ if m= n. (numerator and denominator have same degree)