LIMITS A limit is a calculus property of mathematics, that is, it’s a property of functions that may or may not exist for any given function f(x). A limit is formally written as where x can be any variable present in the function and k is a constant. Informally, I like to write limits as or for simplicity’s sake. Basically, a limit tells you what f(x) is doing when x is around a value k, without actually telling you f(k). A more formal way to state this is “For a function f(x), as x approaches a constant k, f(x) approaches a constant L”. To compute a limit, simply make x get closer and closer to k from both sides – i.e.: make x smaller for x > k and make x bigger for x < k -- without actually becoming it. Should f(x) approach a constant (or function if you’re computing the limit of f(x,y) or some other function with two variables), that constant will be the limit. Some examples of computing limits:
x y x y -1 1 1 1 -0.1 0.01 0.1 0.01 -0.01 0.0001 0.01 0.0001 -0.001 0.000001 0.001 0.000001 As you see, as x gets closer and closer to 0 from both sides, so does the function. Therefore, the limit is 0.
x y x y -1 0.5 1 2 -0.1 0.93 0.1 1.07 -0.01 0.993 0.01 1.006 -0.001 0.9993 0.001 1.0006 Once more, as x gets closer and closer to 0 from both sides, the function gets closer to 1. Therefore the limit is 1.
x y x y -1 -1 1 1 -0.1 -10 0.1 10 -0.01 -100 0.01 100 -0.001 -1000 0.001 1000 As negative values of x get closer to zero, the function gets closer to negative infinity – i.e.: it gets infinitely smaller – whereas when positive values of x get closer to zero, the function get’s closer to positive infinity – it gets infinitely larger. Therefore, there is no limit.
Now, if we wanted to formalize the process, we can say that making values of x for which x < k get closer to k is taking or computing the left-hand limit, whereas making values of x for which x > k get closer to k is taking the right-hand limit. Now that you know about limits with a constant k, you should know that some limits don’t have actual constants as a value of k, but rather made up constants, namely, infinity and negative infinity, which as far as limits are concerned, mean, let x get infinitely bigger and let x get infinitely smaller respectively, and as such, you don’t actually need to take both limits since one of the “hand” limits will be impossible. But now that we’re done with it, let me tell you a secret, for many functions, as long as you don’t end up with division by zero, limits can be discarded in favor of something much quicker. For every continuous f(x) in which f(k) does not cause a division by zero,
Well, basically, a continuous function is a function that has a single curve for a graph, should it have more than a single curve for a graph, it’s continuous within the ranges of x that make the curves. As a rule of thumb:
Every simple polynomial function of a positive, odd degree is continuous all over Every simple polynomial function of a fractional even degree is continuous when x is positive Every simple polynomial function of a negative degree is continuous when x isn’t 0 sin(x) and cos(x) are continuous all over. tg(x) and sec(x) are continuous when cos(x) isn’t 0 csc(x) and cot(x) are continuous when sin(x) isn’t 0 Exponential functions are continuous all over. By extension the hyperbolic trig functions are continuous all over. Logarithmic functions are continuous for every x > 0 Re(x) and Im(x) are continuous all over. The sum, difference and product of these are continuous all over. The quotient is continuous whenever the denominator isn’t 0.
That being said, half the time any given function you’ll find is continuous all over, except for one or two points so don’t bother too much with limits. To close this, some important limits:
Introduction to Limits. Next to be done is a cheat sheet of Limit Properties and then start with derivates. For more information or question...