1. ## Determining limits

I have to determine the limits from the following expressions:

I have the results available but I have no idea what to do, how to do it and why I have to do.
I have searched theses forums and the internet as well as my syllabus for an explanation but have been unable to find one so hope people can help out.

2. ## Re: Determining limits

For this type of problem with rational functions of polynomials, we divide numerator and denominator by the greatest power of $n$ in the expression. This makes those terms tend to a constant while the rest go to zero.

The second example:
\displaystyle \begin{align*} \lim_{n \to \infty} \frac{3n^2 - 4}{-2n^3 +7} &= \lim_{n \to \infty} \frac{\frac{1}{n^3}(3n^2 - 4)}{\frac{1}{n^3}(-2n^3 +7)} \\ &= \lim_{n \to \infty} \frac{\frac3n - \frac4{n^3}}{-2 +\frac7{n^3}} \\ &= \frac{0+0}{-2+0} \\ &= 0 \end{align*}

3. ## Re: Determining limits

Thanks I get it now

4. ## Re: Determining limits

Once you have done that a few times, you should be able to see that:

$\displaystyle \lim_{x\to \infty}\frac{a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0}{b_mx^m+ b_{m-1}x^{m-1}+ \cdot\cdot \cdot+ b_1x+ b_0}$ is

1) $\displaystyle \infty$ if n> m.

2) 0 if n< m.

3) $\displaystyle \frac{a_n}{b_m}$ if m= n