One way of doing this is to express (x^2+7x+11)/(x+5)x2+7x+11x+5 into partial fractions.
Like this : f(x)=(x^2+7x+11)/(x+5) color(red)= (x^2+7x +10-10+11)/(x+5) color(red)= ((x+5)(x+2)+1)/(x+5) color(red)= (cancel((x+5))(x+2))/cancel((x+5))+1/(x+5) color(red)= color(blue)((x+2)+1/(x+5))
Hence f(x) can be written as : x+2+1/(x+5)
From here we can see that the oblique asymptote is the line y=x+2
Why can we conclude so?
Because as x approaches +-oo , the function f tends to behave as the line y=x+2
Look at this : lim_(xrarroo)f(x)=lim_(xrarroo)(x+2+1/(x+5))
And we see that as x becomes larger and larger, 1/(x+5) " tends to " 0
So f(x) tends to x+2 , which is like saying that the function f(x) is trying to behave as the line y=x+2.
