## Special Limits

Recall the definition of a limit: lim_{x—>c} f(x) means

given any positive real number, ε, there
exists a positive real number, δ, such that

0 < |x-c| < δ ==>
|f(x) - L| < ε

### One-Sided Limit: lim_{x—>c+} f(x) = L

means given any positive real number, ε, there
exists a positive real number, δ, such that

0 < x-c < δ ==>
|f(x) - L| < ε

Similarly, lim_{x—>c-} f(x) = L means

given any positive real number, ε, there
exists a positive real number, δ, such that

0 < -(x-c) < δ ==>
|f(x) - L| < ε

### Unbounded Limit: lim_{x—>c} f(x) = ∞

means given any positive real number, M, there
exists a positive real number, δ, such that

0 < |x-c| < δ ==>
f(x) > M

Similarly, lim_{x—>c} f(x) = −∞
means

means given any negative real number, N, there
exists a positive real number, δ, such that

0 < |x-c| < δ ==>
f(x) < M

### Limit at Infinity: lim_{x—>∞}
f(x) = L

means given any positive real number, ε, there
exists a positive real number, M, such that

x > M ==>
|f(x) - L| < ε

Similarly, lim_{x—>-∞} f(x) = L
means given any positive real number, ε, there
exists a negative real number, N, such that

x < N ==>
|f(x) - L| < ε

### Combinations of these Special Limits

A limit can be one-sided *and* unbounded, for example:

lim_{x—>c+} f(x) = ∞

means given any positive real number, M, there
exists a positive real number, δ, such that

0 < x-c < δ ==>
f(x) > M

### Existence of a Limit

If it is stated that, for example, lim_{x—>c} f(x) = L, then this
statement means not only that the limit is L, but that the limit exists.

On the other hand, if it is stated that lim_{x—>c} f(x) = ∞
, then this statement means not only that the limit is unbounded, but that the *limit
does not exist*.

**Examples of ways in which a limit does not exist:**

1. The limit as x approaches c from the right differs from the limit as x
approaches c from the left.

That is, lim_{x—>c+} f(x) ≠ lim_{x—>c-}
f(x)

For example, lim_{x—>0+} |x|/x = 1, and lim_{x—>0-}
|x|/x = -1, so lim_{x—>0} |x|/x does not exist.

2. The limit is unbounded

For example, lim_{x—>c} f(x) = ∞,
then lim_{x—>c} f(x) does not exist.

3. The function oscillates between two fixed values as x approaches c

For example, lim_{x—>0} sin(1/x) does not exist, because
sin(1/x) oscillates between -1 and 1 as x approaches 0.

On the other hand, lim_{x—>0} x sin(1/x) exists, (and can be
proven using the Squeeze Theorem) since although it oscillates faster and
faster as x approaches zero, it does not oscillate between fixed values.

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