What is the formal definition of a limit?

In summary, the formal definition of a limit is that for every value of epsilon greater than 0, there exists a corresponding value of delta greater than 0, such that when the distance between x and a is less than delta, the distance between f(x) and L is less than epsilon. This means that as x approaches a, f(x) gets closer and closer to L. The example given shows how to prove that the limit of 5x - 3 as x tends to 1 is 2 by finding a suitable value of delta based on epsilon.
  • #1
mathshead
can someone explain what the formal difinition of a limit ?
 
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  • #2
n -> [oo] An < +/- [oo]
 
  • #3
the limit as x tends to a of f(x) equals L if;

for every number epsilon (usually, can't find the symbol) > 0, there is a corresponding number [sig] > 0, such that for all x;

0 < |x-a| < [sig] ==> |f(x) - L| < epsilon.

Is the formal definition.
for example (easy one);
To show that the limit of 5x - 3 as x tends to 1 is actually 2;

so a = 1, L = 2 (since this is what it does appear to converge to). Need
0 < |x-1| < [sig] for any epsilon > 0.

f(x) is within epsilon of L ie. |f(x) - 2| < epsilon. So to find [sig] from this,

|(5x-3) - 2| = |5x - 5| < epsilon
5|x - 1| < epsilon
|x - 1| < epsilon / 5.

so [sig] = epsilon / 5

and from 0 < |x - 1| < [sig] = epsilon / 5,

|(5x - 3) - 2| = 5 |x - 1| < 5 (epsilon / 5) = epsilon.

Which proves that L = 2.

Alternatively find a good textbook :wink:
 
  • #4
The limit definition looks rather convulated when stated in terms of epsilons and deltas. One good way of thinking about it is this:

Given any allowable magnitude of error (formally epsilon) from a value (the limit), there exists a range near c (the value x is approaching) for which the function's outputs ( f(x) ) will deviate from the limit no more than the given magnitude of error (epsilon).

The key here is that if the limit for f(x) at a particular point c exists (and hence the previous statement holds), then we are stating that we can get f(x) as close to L as we want. I can make it within .001 or .000001, ... anything (because for each error I present to it, the limit existing garuntees that i can find an interval of values for x symmetrically about c such that f(x) will be that close to L).
 

1. What is the definition of a limit?

The definition of a limit is a fundamental concept in mathematics and describes the behavior of a function as its input approaches a certain value. In other words, it is the value that a function approaches as its input gets closer and closer to a specific value.

2. How is a limit written mathematically?

A limit is typically written using the notation "lim f(x) = L" where f(x) is the function and L is the limit. The notation also includes the variable approaching the limit value. For example, "lim x → 3" means the limit of the function as x approaches 3.

3. What is the difference between a left-hand and right-hand limit?

A left-hand limit is the value that a function approaches as its input approaches a certain value from the left side. A right-hand limit, on the other hand, is the value that a function approaches as its input approaches a certain value from the right side. If the left-hand and right-hand limits are equal, then the overall limit exists.

4. What does it mean for a limit to not exist?

If a limit does not exist, it means that either the left-hand and right-hand limits are not equal, or that one or both of the limits are infinite. In other words, the function does not approach a single, finite value as the input approaches the limit value.

5. How is the definition of a limit used in calculus?

The definition of a limit is a crucial concept in calculus and is used to determine the derivative and integral of a function. It is also used to analyze the behavior of functions and make predictions about their values. Without the concept of a limit, many fundamental principles in calculus would not be possible.

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