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existence and value of the limit of a function

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I need help on these two problems please

1. Does the existence and value of the limit of a function f(x) as x approaches x_o ever depend on what happens at x = x_o? Explain and give examples.

2. What does it mean for a function to be continuous? Give examples to illustrate the fact that a function that is not continuous on its entire domain may still be continuous on selected intervals within the domain.

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The existence and value of the limit of a function are depicted.

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1. Does the existence and value of the limit of a function f(x) as x approaches x_o ever depend on what happens at x = x_o? Explain and give examples.

Solution:

The definition of limit is

Let f be a function from the real numbers (R) to the real numbers (R), and let x be a real number. Lim(x->x ) f(x) = L which means that for every > 0, there is a > 0, such that if > |x - x | > 0, then > |f(x) - L|.

From the above definition it follows that the existence and value of the limit of a ...

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