Limits Confusion

I would just like some clarity on a question that has been dwelling in my mind lately.

theres a limit, $\lim\limits_{x \to 1 }x$
The value of this limit no matter what x is, is somehow always 1?
could any one explain this to me.

thank you!!!

Solution 1:

The easy way to view this is that, whatever you write next to the word "$\lim$" will be a function that you are referring to. The part where you say "$x \rightarrow a$ " is where your function is approaching (or, graphically speaking, the y-coordinate) when your input approaches that specific value of a. Looking at your example: $$\lim_{x \to 1} x = 1$$

The value of this limit no matter what x is, is somehow always 1?

The "$x$" that you are referring to is your function, not an input. Treat it as $f(x)=x$. Now let's look at that graph: Graph of f(x)=x

You can see that as the $x$ approaches 1, your function gets to, or in this case, returns the output of 1. That is why the limit in this case is equal to 1. Now if you change the "$a$" value (as in $x \to a$), the limit will be difference. For example, $$\lim_{x \to 2} x = 2$$ $$\lim_{x \to 5} x = 5$$

Solution 2:

When we do a limit, we do not focus on the value of the expression at any value of $x$. We only focus on the value of the limit at a certain point, which is x = 1.

We use limits to say "What is y for y = x and x = 1?"

So for

$\lim\limits_{x \to ∞ }$($1$ + $1$/$x$$)^x$ = $e$

We ask "What is y = $($1$ + $1$/$x$)^x$ for x = ∞, but we cannot do x = ∞, so we use the limit expression.

Limits are used to express questions like the one above, especially when we use values that cannot equal $x$, such as infinity.

Hope this helps.