Limits
we say that the limit of f(x) as x approaches a is L written lim f(x) = L x–>aif as x gets closer and closer to a without touching it, f(x) gets closer and closer to L
In mathematics, a limit is a concept that describes the behavior of a function as its input approaches a certain value. The limit of a function is the value that the function approaches as its input approaches a particular value.
Formally, we say that the limit of a function f(x) as x approaches a is L, denoted by:
lim f(x) = L, x → a
if we can make the values of f(x) arbitrarily close to L by taking x sufficiently close to a (on either side of a), without actually reaching a itself.
To understand this concept better, consider the following example:
Suppose we have the function f(x) = (x^2 – 4)/(x – 2) and we want to find the limit of this function as x approaches 2.
If we try to directly substitute x = 2 into the expression for f(x), we get an expression of the form 0/0, which is undefined. However, if we look at the behavior of f(x) as x approaches 2 from both sides, we see that:
lim f(x) = 4, x → 2
This means that as x gets closer and closer to 2, the values of f(x) get closer and closer to 4, but never actually reach 4 at x = 2.
Limits are important in calculus because they help us understand the behavior of functions and the properties of curves in the limit. They are used extensively in the study of derivatives, integrals, series, and other mathematical concepts.
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