end behavior of x^2+5/7-2x^2

Since the highest power polynomial in the numerator = the highest power polynomial in the denominator, we have a  "same/same" situation. And the end behavior  - in both directions from the y axis - is determined by the ratio of the coefficients on these "greatest power" variables = 1/-2  = -1/2.

If you want to do this by limits: Need to solve   lim(x→∞) (x²+5)/(7-2x²)   and    lim(x→-∞) (x²+5)/(7-2x²)

Looking at the expression:  (x²+5)/(7-2x²)    divide both the numerator and denominator by x² to get:

     (x²/x² + 5/x²)/(7/x² - 2x²/x²)  =  (1 + 5/x²)/(7/x² - 2)

Now as  x→∞,  lim(1 + 5/x²)/(7/x² - 2) → (1)/(-2)  =  -1/2

    (because as x→∞, both 5/x²→0 and 7/x²→0)

And, as x→-∞ lim(1 + 5/x²)/(7/x² - 2) =  -1/2 also.

So the end behavior is that the funtion approaches -1/2 both to the right and to the left.