Evaluating Limits Algebraically, Part 1 - Problem 1
To evaluate the limit of a continuous function at a certain point, simply evaluate the function at that point. From the definition of continuity, we know that the limit of a function is the same as the value of the function at that point, so we can use this fact to find limits. This is often the easiest way to evaluate a limit, but it is important to check that the function is actually continuous at the point in question.
Let's evaluate another limit. I have the limit as x approaches 1 of x² minus 16 minus x² plus 4x. Now I want to try to use continuity wherever possible, because this is the easiest way to evaluate a limit.
Now using continuity means just plugging the number 1 into the function. You can only do that is the function is continuous at x equals 1. So let's figure out where this function is continuous.
It's a rational function. Rational functions are continuous everywhere that they're defined, but they're undefined when the denominator is 0. So this function is continuous except at x equals 0, and x equals -4.
0, and -4 are the two zeros of the denominator. So that's where this function will fail to be continuous. 1 isn't one of those numbers, so I can plug 1 in. I get 1² minus 16 over 1² plus 4 times 1. So that's 1 minus 16, -15 over 1 plus 4, 5, -3.
So using continuity is the easiest way to evaluate a limit. You always want to do it if you can. You won't always be able to do it, but you should always check. Just make sure that your function is actually continuous at the number in question.