In general, if lim x->a m(x) does not exist and
lim x->a n(x) does not exist, is it true that
lim x->a [m(x)+n(x)] does not exist? How about
lim x->a [m(x)*n(x)]
To determine whether a limit exists or not, we need to consider the individual limits as well as the behavior of the combined expressions.
First, let's address the sum of the functions: lim x->a [m(x) + n(x)].
If both lim x->a m(x) and lim x->a n(x) do not exist, it does not necessarily mean that lim x->a [m(x) + n(x)] does not exist. It is possible for the sum to have a limit even if the individual functions do not. This is because the behavior of m(x) and n(x) around 'a' may compensate each other and lead to a well-defined limit.
For example, consider the functions m(x) = sin(x) and n(x) = -sin(x). Individually, lim x->0 sin(x) and lim x->0 -sin(x) do not exist. However, the sum m(x) + n(x) = sin(x) - sin(x) = 0 has a well-defined limit of 0 as x approaches 0.
Now, let's move on to the product of the functions: lim x->a [m(x) * n(x)].
If either lim x->a m(x) or lim x->a n(x) does not exist, then lim x->a [m(x) * n(x)] also does not exist. This is because in order for the product to have a limit, both m(x) and n(x) must have well-defined limits at 'a'.
To understand why this is the case, consider a scenario where lim x->a m(x) does not exist, but lim x->a n(x) does exist (or vice versa). In that case, let's assume we have a well-defined limit for n(x) at 'a', but m(x) has oscillations or diverges as x approaches 'a'. When multiplying the two functions, the oscillations or divergence of m(x) will affect the behavior of the product, preventing it from having a well-defined limit.
In summary, if both lim x->a m(x) and lim x->a n(x) do not exist, it is possible for lim x->a [m(x) + n(x)] to exist. However, lim x->a [m(x) * n(x)] will not exist if either lim x->a m(x) or lim x->a n(x) does not exist.