# Anyone can explain to me what's the difference between "limit", "limsup" and "liminf" of a function? It would be helpful to explain with concrete example.

I'll try to give an example below.

Example 1:

Second example:

To obtain the outcome, we can apply the squeeze theorems on the left and right.

The value lim inf = lim sup indicates that the limit is also that value.

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The limit of a function represents the value that the function approaches as the input approaches a certain value. The limsup (limit superior) and liminf (limit inferior) are used to describe the behavior of a function when it oscillates or fluctuates near a certain point.

The limsup of a function is the largest limit point that the function can reach as the input approaches a certain value. It is the supremum (or maximum) of all the limit points.

The liminf of a function is the smallest limit point that the function can reach as the input approaches a certain value. It is the infimum (or minimum) of all the limit points.

To illustrate with an example, let's consider the function f(x) = sin(1/x). As x approaches 0, this function oscillates between -1 and 1 infinitely many times. The limit of f(x) as x approaches 0 does not exist because the function does not approach a single value.

However, we can find the limsup and liminf of f(x) as x approaches 0. The limsup is 1, as the function reaches its maximum value of 1 infinitely many times. The liminf is -1, as the function reaches its minimum value of -1 infinitely many times.

In summary, the limit of a function represents the value it approaches, while the limsup and liminf describe the largest and smallest limit points respectively, when a function oscillates or fluctuates near a certain 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.

- How do you find the limit #lim 5^(x^2-2x+1)# as #x->3#?
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- How did earlier mathematicians calculate limits so accurately?
- For what values of x, if any, does #f(x) = cot(pi/8-2x) # have vertical asymptotes?
- How do you find the limit of #(x - (1/x))^2# as x approaches infinity?

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