On epsilon delta definition fo Limit.

Question

Is it possible to use $\delta=\min(1,\frac\varepsilon c)$ in the following exercise? Thanks in advance!!

$$\lim_{x\to 1}(x^2+4x)=5.$$

To make my question clear, do we have a right to choose the limit constant a as $\delta$?

Answer 1

For any $ \epsilon >0$, let $\delta = \min\{1, \frac{\epsilon}{7}\}$. Then when $|x-1| < \delta$, clearly, $|x+5|=|x-1+6|<|x-1|+6 \le 7$, and hence

$$|x^2+4x-5|=|(x+5)| \times |(x-1)| \le 7\delta \le \epsilon.$$