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 Math Help > Calculus > Limit > Special Limits

## Special Limits

Recall the definition of a limit:  limx—>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: limx—>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, limx—>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: limx—>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, limx—>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: limx—>∞ 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, limx—>-∞ 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:

limx—>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, limx—>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 limx—>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, limx—>c+ f(x) ≠ limx—>c- f(x)
For example, limx—>0+ |x|/x = 1, and limx—>0- |x|/x = -1, so limx—>0 |x|/x does not exist.

2. The limit is unbounded

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

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

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

On the other hand, limx—>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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The webmaster and author of this Math Help site is Graeme McRae.