This is how I was taught to do it without L'Hospitals rule
The trick that I used is to observe all the terms as they approach a number that is getting infinitely larger. Since it's specified we only want it as it approaches positive infinity we don't need to observe it as it approaches a number that gets infinitely smaller, since the limit from the left and the right as it approaches positive infinity will be infinitely large.
So first I look at the top and see that 5 is raised to X. What happens to 5 as it's exponent gets infinitely larger? The result is infinitely larger, so I just substituted it with a + sign to remind myself it will stay positive. It now looks like:
Then I observe the bottom. The limit of a constant is the constant, so we know as X gets infinitely larger 1 will stay 1.
The interesting behavior is at
. As x gets infinitely larger, it stays positive. You can test this by grabbing any calculator and taking the natural log of any extremely large positive number.
What is 1 subtract a large positive number? A negative of course, which gives me (using my symbology) -
which comes out to be a negative. This tells me that the limit will approach negative infinity due to the behavior in the denominator.
This trick works for simple stuff like this. Even if you won't learn L'Hospital's rule in your class until second semester calc, you should pick it up if not just to check your work.