Can a limit have 2 values?

Question

Limit of a function is said to have only one value.

But say I have a function

$$\lim_{l\to \infty}\left(1+\frac 1{{(1+x^2)}^l}\right)$$

Is this limit not defined or is a function of $x$?

Answer 1

There are several related but separate notions.

If you consider $x$ as standing for some fixed (but arbitrary) real number then you can consider the questions what is the limit of $$\lim_{l\to \infty}\left(1+\frac 1{{(1+x^2)}^l}\right)$$ for this one fixed $x$.

The limit and whether it exists will then usually depend on $x$, you could denote it $L_x$ if it exists. (If it exists it is unique for a fixed $x$, for example for $x=2$ there is just one limit; in your context, real numbers, a limit is always unique. There is a more general theory of spaces, called topological spaces, where one can consider limits and in some of those limits are not unique. But that's several years beyond what seems to be your current level.)

You can then defined a function, $L$, defined as $x \mapsto L_x$ for each $x$ for which the limit exists.

This $L$ then can be seen as the limit (function) of the functions $f_l(x) = \left(1+\frac 1{{(1+x^2)}^l}\right)$ as $l \to \infty$.

However, there are different notions of the limit of a function. The one above is that of pointwise limit.

Answer 2

The limit is a function of $x$. In this case, if $x=0$ the value is $2$ independent of $l$, so the limit is $2$. If $x \neq 0$ the fraction will go to $0$ and the limit will be $1$.