Suppose lim x→1 (sqrt(ax+b)-2)/(x-1)=1, then a-b = ?
The answer key solved this by seeing the limit as 0/0, and therefore, sqrt(ax+b)-2 as 0.
What's the assumption, sqrt(ax+b)-2=0, based on?
Thanks in advance!
If you substitute the value $\displaystyle x=1$ in the limit then the denominator is equal to $\displaystyle 0$. This means the numerator can't be equal to some $\displaystyle L \in \mathbb{R}\setminus \{0\}$ as the limitvalue would be infinite then (i.e it would not exist). Hence the only possibility is that the numerator is equal to $\displaystyle 0$ as $\displaystyle \frac{0}{0}$ is undefined.