Overview
A sequence {% s_1,s_2,... %} of numbers is said to converge to a lime if there is a number {% L %} that the sequences get closer and closer to, tending to the numbers as the number of elements of the sequence goes out to infinity.
As an example, the sequence of numbers {% 1,\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, ... \frac{1}{2^n},... %} tends to the number {% 0 %}.
Definition
A sequence {% s_1,s_2,... %} has a limit {% L %} if, for every positive numbr {% \epsilon %}, there is a positive number {% N %} such that
{% |s_n - L| < \epsilon %}
whenever n > N.
Cauchy Sequence
A sequence {% s_1,s_2,... %} in a metric space is Cauchy if, for every positive numbr {% \epsilon %}, there is a positive number N such that
{% |s_n - s_m| < \epsilon %}
whenever n, m > N.
- Any convergent sequence is Cauchy
- A space where every Cauchy sequence converges is called complete. That is, a complete space contains its limit points.
Limit Theorems
Let {% f %} and {% g %} be functions such that
{% \lim_{x \to p} f(x) = A %}
and
{% \lim_{x \to p} g(x) = B %}
Then the following holds.
- {% \displaystyle \lim_{x \to p} f(x) + g(x) = A + B %}
- {% \displaystyle \lim_{x \to p} f(x) - g(x) = A - B %}
- {% \displaystyle \lim_{x \to p} f(x) \times g(x) = A \times B %}
- {% \displaystyle \lim_{x \to p} f(x) / g(x) = A / B %}