# Limit of a function

The **limit of function f as x approaches c is L** if

*f(x)*can be made to be as close to

*L*as desired by making

*x*sufficiently close to

*c*:

Or:

For example:

because:

You can’t always find the same limit when you approach from both sides, that’s why we define the **lateral or one-side limits**:

–The **limit of a function f as x approaches a from the left is L**

^{-}if f(x) can be made to be as close to L

^{-}as desired by making x sufficiently close to

*a*from below:

–The** limit of a function f as x approaches a from the right is L^{+}** if f(x) can be made to be as close to L

^{+}as desired by making x sufficiently close to

*a*from above:

Then, in the example

Other definitions:

NOTE: Remember that when

we have a vertical asymptote

*Demonstration*: Imagine we have two limits: b and c and b≠c. Then

If we choose ε as in the picture there is a contradiction in b using δ_{1}. Then b = c and the limit is unique.

**Exercise**: calculate the limit of *f* as *x* approaches 0 and 2, if:

Solutions

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