calculus - Methods for Finding Limits as x Approaches Infinity

I have a question about the best method to find the limit of a function as it approaches infinity.

The limit as x approaches infinity of (x^3+5x)/(2 x^3-x^2+4) is 1/2.

I found this just by taking the largest values of x (X^3/2x^3) and plugging in a number (1). I proceeded under the assumption that if x gets large it will subsume the lesser values of x and the constants, so in a way these lesser values are not necessary.

The book I am using (and other posts on this site), however, provide a different method (i.e., divide the numerator and the denominator by the largest value of x, and then use the limit laws to find the result). This method is obviously much more cumbersome, but I am not sure if any rigor is gained by it.

My question is: Is there a case where the first method will produce a wrong answer? Put another way, will the limit as x approaches infinity of (ax^2+bx+c/dx^2+ex+f) always be equal to the limit as x approaches infinity of (ax^2/dx^2)?