Answer
$-\frac{2}{9}$
Work Step by Step
We have $\lim _{x\rightarrow -1}\frac {x^{2/9}-1}{x+1}$.
Similarly, let's rewrite the expression as $\lim _{x\rightarrow -1}\frac {f(x)-f(-1)}{x-(-1)}$, where $f(x) = x^{2/9}$.
Again, this is the definition of the derivative of $f(x)$ at $x=-1$, so we have:
$\lim _{x\rightarrow -1}\frac {x^{2/9}-1}{x+1} = f'(-1)$.
Taking the derivative of $f(x) = x^{2/9}$, we get $f'(x) = \frac{2}{9}x^{-7/9}$.
Evaluating $f'(-1)$ gives us $\frac{2}{9}(-1)^{-7/9} = -\frac{2}{9}$.
Therefore, the limit is $-\frac{2}{9}$.
