Question

How do you find the limit of the cubic root of x^2+1 all mines x ? Thank you.

Answer 1

The question does not say what number `x` is approaching.

Explanation:

The properties of limits assure us that for any number `a`,

`lim_(xrarra)(root(3)(x^2+1)-x)` can be found by substituting `a` in place of `x` and doing the arithmetic.

For any `a`, the subtraction property of limits gets us:

`lim_(xrarra)(root(3)(x^2+1)-x) = lim_(xrarra)root(3)(x^2+1)-lim_(xrarra)x`

The power (or root) property of limits lets us continue:

`= root(3)(lim_(xrarra)(x^2+1))-lim_(xrarra)x`

Using the addition property:

`= root(3)(lim_(xrarra)(x^2)+lim_(xrarra)1)-lim_(xrarra)x`

And the power property again

`= root(3)((lim_(xrarra)x)^2)+lim_(xrarra)1)-lim_(xrarra)x`

Now use the fact that `lim_(xrarra)x = a` to write

`= root(3)(a^2+1)-a`