Evaluating Limits Algebraically, Part 1 - Problem 3
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.
We're evaluating limits algebraically. Now when you're evaluating limits, remember sometimes you can use continuity, and sometimes you can. Let's see if we can use continuity to evaluate this limit. Limit is x approaches of 2x minus 50 over root x minus 5.
Now where is this function continuous? Well, it's a combination of continuous functions, so it's continuous wherever it's defined. It's going to be defined for this radical x greater than or equal to 0. To keep 0 out of the denominator, I'm going to need this root x not to be 5. So I need x not to be 25.
So as long as x is greater than or equal to 0, and not 25, this function is defined, and continuous. So 9 is okay. That means I can use continuity to evaluate this limit. Let me plug 9 in for x. I get 2 times 9 minus 50 over root 9 minus 5. This is 18 minus 50 over 3 minus 5. 18 minus 50 is -32. 3 minus 5 is -2. That's 16. So the limit is x approaches 9 of this function is 16.