Question

Find the oblique or (slant asymptote) of y = (5x^3 +6x^2) / (5x^2 + 4x + 5) and express your answer in the form y = mx + b, where m and b are constants

Answer 1

I've done oblique asyptotes years ago. Maybe I can help you out, there are several steps that you must take.

1) Check if your function is an oblique asyptote my finding the limi as x --> infinity
2) If you end up with infinity, then youhave an oblique asymptote
3) After that divide your function y/x to find m, since y = mx + b; mx = y, m = y/x. Then simplify your expression.
4) Then after finding y/x, which is the slope, subtract it from the original mx to find b , so b = (y - mx) . Remember mx = the new slope that you have found.
5) finally you have the slope + b, so you can then write the equation of the OA.

Those were general steps that can help you solve any problem that has an OA, as for your question:


\[y = 5x^3 + 6x^2/ 5x^2 + 4x + 5\] , step number (1) divide it by x and you'll get
\[y = (5x^3 + 6x^2)/ x(5x^2 + 4x + 5)\]
multiply x using the distributive law and you will have :
\[y = 5x^3 + 6x^2 /5x^3 + 4x^2 + 5x\]

now find the limit of this as it goes to infinity:
m = \[\lim_{x \rightarrow \infty} = 5x^3/5x^3\] = 1

After that we need to find b now, we know that b = y - mx, and we have found m = 1, so m = x.

Now subtract y ( the original function) from m ( which is x):

\[(5x^3+6x^2 / 5x^2 + 4x+ 5 ) - x\]

do the simplification and you'll end up with :
\[5x^3+ 6x^2 - 5x^3 - 4x^2- 5x/5x^3 + 4x^2 + 5x\]
= \[2x^2-5x/ 5x^2 + 4x + 5\]

then find the limit of that as x goes to infinity:

\[\lim_{x \rightarrow \infty} 2x^2/5x^2 = 2/5\]

Finally write your equation in a more decent form:

- m = 1
-b = 2/5

so y = x + 2/5

Done :)