## University Calculus: Early Transcendentals (3rd Edition)

$$\lim_{x\to1}\frac{x^{50}-1}{x-1}=50$$
$$A=\lim_{x\to1}\frac{x^{50}-1}{x-1}=\lim_{x\to1}\frac{x^{50}-1^{50}}{x-1}$$ According to the definition of derivative, we know that $$\lim_{x\to z}\frac{f(x)-f(z)}{x-z}=f'(z)$$ So here if we take $f(x)=x^{50}$, we have $$f'(z)=\lim_{x\to z}\frac{x^{50}-z^{50}}{x-z}=\frac{d(z^{50})}{dz}=50z^{49}$$ Take $z=1$, then $$f'(1)=\lim_{x\to1}\frac{x^{50}-1^{50}}{x-1}=50\times1^{49}=50$$ $$A=50$$