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Evaluating Limits Algebraically, Part 1 - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
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.

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