Answer
$ 50$
Work Step by Step
To evaluate the limits by first converting each to a derivative at a particular x-value, we can use the concept of derivatives and apply L'Hôpital's rule.
We have $\lim _{x\rightarrow 1}\frac {x^{50}-1}{x-1}$.
Let's rewrite the expression as $\lim _{x\rightarrow 1}\frac {f(x)-f(1)}{x-1}$, where $f(x) = x^{50}$.
Now, we can recognize this as the definition of the derivative of $f(x)$ at $x=1$, so we have:
$\lim _{x\rightarrow 1}\frac {x^{50}-1}{x-1} = f'(1)$.
Taking the derivative of $f(x) = x^{50}$, we get $f'(x) = 50x^{49}$.
Evaluating $f'(1)$ gives us $50(1)^{49} = 50$.
Therefore, the limit is 50.
