# What is the definition of limit in calculus?

There are several ways of stating the definition of the limit of a function. In order for an alternative to be acceptable it must give the same results as the other accepted definitions. Those other definitions are accepted exactly because they do give the same results.

The definition of the limit of a function given in textbooks used for Calculus I in the U.S. is some version of:

if and only if

That is the end of the definition

Comments Tlhe following version is a bit more "wordy", but it is clearer to many.

for which the following is true:

An acceptable rephrasing of that "if . . ., tlhen . . . " is:

The Game

What I am claiming is that:

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In calculus, the limit is a fundamental concept that describes the behavior of a function as the input approaches a certain value. It represents the value that the function approaches or tends to as the input gets arbitrarily close to the specified value. The limit can be thought of as the instantaneous value or the value that the function would approach if it were to be evaluated at that specific point.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the limit as x approaches 0 of #(1-cos(x))/sin(x)#?
- How do you find the limit of #f(x) = (x^2 + x - 6) / (x + 3) # as x approaches 2?
- How do you find the limit of #arctan(x)# as x approaches #oo#?
- How do you evaluate the limit #sin(5x)/x# as x approaches #0#?
- How do you evaluate the limit #sqrt(x-2)-sqrtx# as x approaches #oo#?

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