PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
To find the limit of a rational function as x approaches a point, first determine where the function is continuous. If it is continuous at the point that x is approaching, then you can simply evaluate the function at that point because by the definition of continuity, the value of the function at a point should be the same as the limit of the function at that point.
Let's take a look at another example. I want to evaluate the limit as x approaches -5 of x over x plus 4 plus 3 over x plus 3. Now I want to see if I can use continuity on this limit. So I need to determine where this function is actually continuous.
Now this function is a combination of functions I know to be continuous. I've got little rational function here plus 3. So the sum of continuous functions. X plus 3 is a polynomial, and quotients of continuous functions are continuous. So this is going to be continuous wherever its defined.
Where is it defined? Well, it's not defined at -3, so I would have to say continuous except x equals -3. It's also not defined at -4. But -5 is okay, so I can use continuity to evaluate this limit. That means plugging -5 in for x.
So I get -5 over -5 plus 4, -1 plus 3 all over -5 plus 3. So 5 plus 3, that's 8 over -5 plus 3, -2. I get -4. Very easy to evaluate a limit using continuity.