Efficient ways to determine if a function is continuous or not?

[GrannySmith]

I am taking functions trigonometry this year. Right now, we are learning about continuous and discontinuous functions.

Algebraically, how would you determine continuity of a function efficiently? For example lets say : √(x^2 -4); at x=-5

They only give you this algebraically. No graphs, no tables, nothing. I understand the concept of continuous functions. Basically, f(x) needs to approach 5 from the left and the right.

What I don't understand, is how you determine this without a calculator. In my textbook, they make an x/f(x) table. Then you need to determine the value of f(x) when x = -5.1,-5.01,-5.001 and then at x= -4.9,-4.99,and -4.999. Plugging these values in takes forever, and I feel like its not efficient.

Is there another method to doing these kinds of problems, or is my calculator the only option? I'm also not sure if calculators are allowed in functions trig, and don't want to rely on them unless absolutely needed.

Any help would be greatly appreciated! Thanks!

 

[HallsofIvy]

"Plugging in numbers" will not tell you anything about continuity. There are infinitely many functions that are continuous right up to, say, x= 1.99999999999999, and then change sharply and are NOT continuous at x= 2.

To determine the continuity of complicated functions you need to use knowledge about the continuity of simple functions from which it is built and such theorems as "if f(x) and g(x) are both continuous at x= a then so afe f+ g and fg" and "if f(x) is continuous at x= a and g(x) is continuous at f(a) then g(f(x)) is continuous at x= a".

Here, you are given f(x)= √(x^2- 4). That can be written as f(x)= h(g(x)) with h(x)= √x and g(x)= x^2- 4. Now, you should know that the square root function is continuous for all non-negative x and any polynomial is continuous for all x. So f(x) is continuous as long as x^2- 4>= 0. And, as you could see by looking at a graph of y= x^2- 4, that is true for x>= 2 and for x<= -2.
The function, f(x)= √(x^2- 2) is only defined for x<= -2 and x>= 2 and is continuous wherever it is defined.

(You will see that a lot of commonly used functions are "continuous wherever defined". There is no deep mathematical property here, just that continuity is so useful that our ways of writing functions has grown up to make it particularly easy to write continuous functions. In a very precise sense, "almost all" functions are never continuous.)

Basically, f(x) needs to approach 5 from the left and the right.

NO- although you may just have said it wrong. f(-5)= √(25- 4)= √21 In order that f be continuous at x= -5, f(x) needs to approach √21 as x approaches -5 from the left and right.