How do I solve this limit problem?

[temporaryinsanit]

What steps do I take to get an answer for these two limit problems? Can someone tell me how to get a solution, and all the steps taken to get it. Thank You

A. lim x² + 2x – 3 / x³ + 2x²
x?-?

B. lim e^x / 1 - X³
x??

 

[galactus]

The first one is rather intuitive if you note that the power of the denominator is larger than the power of the numerator.

Discard all terms except \(\displaystyle \lim_{x\to -\infty}\frac{x^{2}}{x^{3}}\) and what is the limit?.

Check out the second one is a similar manner. e^x expands faster than x^3.

 

[soroban]

HGelolo, temporaryinsanit!

\(\displaystyle (A)\;\;\lim_{x\to-\infty}\frac{x^2+2x-3}{x^3+2x^2}\)

If you must show your work, here are the steps . . .


\(\displaystyle \text{Divide top and bottom by }x^3\!:\quad \lim_{x\to-\infty}\frac{\frac{x^2}{x^3} + \frac{2x}{x^3} - \frac{3}{x^3}} {\frac{x^3}{x^3} + \frac{2x^2}{x^3}}\)


\(\displaystyle \text{And we have: }\;\;\lim_{x\to-\infty}\frac{\frac{1}{x} + \frac{2}{x^2} - \frac{3}{x^3}} {1 + \frac{2}{x}} \;=\; \frac{0+0-0}{1+0} \;=\;\frac{0}{1} \;=\;0\)