What is the definition of limit in calculus?

Answer 1

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:

Definition Let #f# be a function defined on some open interval containing #a# (except possibly at #a#). Then the limit as #x# approaches #a# of #f# is #L#, written:
#color(white)"ssssssssss"# #lim_(xrarra)f(x)=L#

if and only if

for every #epsilon > 0# there is a #delta > 0# for which:
if #0 < abs(x-a) < delta#, then #abs(f(x) - L) < epsilon#.

That is the end of the definition

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

for every #epsilon > 0# (for every positive epsilon), there is a #delta > 0# (there is a positive delta)

for which the following is true:

if #x# is any number for which #0 < abs(x-a) < delta# is true, then #abs(f(x) - L) < epsilon# is also true.

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

If #x# is a chosen number within distance #delta# of #a# (but #x!=a# because weird stuff might happen right at #a#),
then #f(x)# is a number within distance #epsilon# of #L#

The Game

I claim that #lim_(xrarra) f(x) = L#.

What I am claiming is that:

if someone else chooses how close I need to make #f(x)# to #L# (give me a distance #epsilon#)
then I'll show that if you start with an #x# close enough to #a# (within #delta# of #a#) then you'll get an #f(x)# within #epsilon# of #L#. Because I am only making a claim about values of #f(x)# for #x#'s chose to #a#, the #x# chosen cannot be equal to #a#. (For nice functions this last bit about #x=a# won't matter, but not all functions are nice.)
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Answer 2

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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Answer from HIX Tutor

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.

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.

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