Question

What is `\lim _ { x \rightarrow - 3^ { + } } \frac { 2x } { x + 3}`?

Answer 1

The limit is `-oo`. See explanation.

Explanation:

If we substitute `x=-3` to the expression we get:

`lim_{x->3^+}(2x)/(x+3)=(-6)/0`

We got zero in the denominator, so the resulting limit is either (`+oo`) or (`-oo`).

To determine the sign of the limit we have to find the signs of both numerator and denominator.

The numerator is `(-6)` so it is negative. To check the sign of the denominator we have to substitute a proper value different than `-3`. We calculate the right-hand side limit (which is indicated by the + sign next to the number `-3`), so we have to substitute a value bigger than -3 (e. g. -2).

If we do so, the denominator becomes `-2+3=1 >0`. This means that the denominator goes to zero through positive values:

`lim_{x->3^+}(2x)/(x+3)=(-6)/0^+`

The numerator and denominator are of different signs, so their quotient is negative, so the limit is `-oo` (minus infinity)