Question

Prove `lim_(xrarr5)(2x+4)=14`?

Answer 1

We can prove this in a number of ways. Since this is classified as pre-calculus, I will avoid an `epsilon - delta` proof.

In general, a limit is the value that a function approaches. Let
`f(x) = 2x+4`.

First, let's just justify it numerically.

If we plug in values just barely less than `x=5`, we see that the values approach 14, i.e.
`f(4) = 12`
`f(4.9) = 13.8`
`f(4.99) = 13.98`
`f(4.999) = 13.998`
and so on and so forth

The same applies if we plug in values just barely above `x=4`:
`f(6) = 16`
`f(5.1) = 14.2`
`f(5.01) = 14.02`
`f(5.001) = 14.002`

This means that as `x rightarrow 5`, the function approaches 14.

Now, we can also sort of prove this limit:

For a function that is "simple", such as this line, we can actually just plug in the value and get our answer, too.

`f(5) = 2 * 5 + 4 = 14`