In this video,
I want to list the main tools that we can use to compute limits of sequences.
I am not going to start from scratch.
I will assume that you are already familiar with...
...the most important properties of limits of functions.
Limits of sequences behave in a similar way.
I will explain which things work exactly the same.
(And I won’t try to write the precise statement of each theorem or its proof.)
And then, I will also point out the few differences.
What works the same to compute limits of sequences versus limits of functions?
Probably the most important tool to compute limits of functions are...
...the limit laws.
There are identical theorems for limits of sequences:
same theorems, same proofs, same applications.
Next, we have the Squeeze Theorem.
There is also a version of the Squeeze Theorem for sequences:
same theorem, same proof, useful for similar examples.
The third important tool to compute limits of functions is L’Hôpital’s Rule.
This one, unfortunately, does not extend to sequences.
We cannot take the derivative of a sequence!
For example, what is the derivative of n!?
However, in some cases,
even if I start with a sequence (which has domain of only the natural numbers),
I can extend it to a function defined over all of the reals,
and then I can use L’Hôpital’s Rule for that function.
But even in that case, I have to be careful.
Let’s say that I begin with a function f
with domain of all the real numbers, starting at an integer c.
I can consider the sequence {aₙ},
which is the restriction of f, to just natural inputs.
For example, this could be the graph of the function, in blue.
And in red, I have the graph of the sequence.
In this case, what's the relation between...
...the limit of the function and the limit of the sequence?
Well in this situation,
IF the limit as x approaches ∞ of f(x) exists and is a real number L,
THEN the limit of the sequence is ALSO L.
IF the lim_(x → ∞) f(x) = ∞,
THEN the limit of the sequence is ALSO ∞.
But, IF the limit of the function does not exist for a different reason,
(wild oscillation)
THEN the sequence may or may not have a limit.
I cannot draw a conclusion!
This is best illustrated with an example.
Here is the graph of a function, in blue.
This function does not have a limit as x approaches ∞,
because it keeps oscillating between 0 and 1.
However, the corresponding sequence happens to be constant,
and it does have a limit!
To conclude the video, what happens when I compose a function and a sequence?
For example, what is the limit, as n approaches ∞,
of e^(1/n!)?
I can think of this sequence as the composition of the exponential function,
and the sequence {1/n!}.
We have a theorem for this situation.
Let {aₙ} be a sequence, and let f be a function.
I would like to say that IF aₙ converges to the real number L,
THEN the sequence f(aₙ) also converges to f(L).
But I need a bit more.
For this to work, I need to require the function f to be CONTINUOUS.
This theorem should not a surprise:
it has a counterpart for the limit of a composition of two functions.
It works the same way and has a similar proof.
I invite you to try to write the proof of this theorem as an exercise.
It is a great exercise,
because you have to use BOTH the definition of limit of a function,
and the definition of limit of a sequence.
Now I can go back to the example above and compute that limit.
The sequence {1/n!} has limit 0,
and the exponential function is continuous;
(and in particular it's continuous at 0)
therefore, the limit of the sequence e^(1/n!),
is e⁰, which is 1.
In summary, everything you knew about computing the limit of a function...
...as the variable approaches ∞,
works exactly the same way when computing the limit of a sequence.
Same theorems, same proofs.
The only exception is L’Hôpital’s Rule,
because we cannot compute the derivative of a sequence,
unless it comes from a function with domain of the positive reals.