# How do you use the limit definition to prove a limit exists?

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The definition of limit of a sequence is:

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To use the limit definition to prove a limit exists, you need to show that for any given epsilon (ε) greater than zero, there exists a corresponding delta (δ) greater than zero such that if the distance between the input value (x) and the limit value (L) is less than delta, then the distance between the function value (f(x)) and the limit value (L) is less than epsilon. In other words, you need to demonstrate that as x approaches a certain value, f(x) approaches a specific limit value. This can be done by manipulating the limit definition equation and finding a suitable expression for delta in terms of epsilon.

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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 of #| x - 5 |# as x approaches #5#?
- How do you prove that the limit of #((x^2+x-6)/(x-2))=5# as x approaches 2 using the epsilon delta proof?
- What is the limit of #Tan(x)# as x approaches infinity?
- What is #lim_(x->oo) x - sqrt(x^2 + 4x + 3)#?
- How do you evaluate the limit #((3+h)^3-27)/h# as h approaches #0#?

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