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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The webmaster and author of this Math Help site is Graeme McRae.