$$\lim_{x\to2} \frac{e^{x^2}-e^{2x}}{(x-2)e^{2x}}$$ I tried solving this limit using some standard limit methods but could not come to a successful conclusion.
I would be highly obliged if the respected community provides me with some help. Thank you!
Note that : $$\frac{e^{x^2}-e^{2x}}{(x-2)e^{2x}}=\frac{e^{x^2-2x}-1}{x-2} $$ And therfore: $$\lim_{x\to2} \frac{e^{x^2}-e^{2x}}{(x-2)e^{2x}}=\lim_{x\to2} \frac{e^{x^2-2x}-1}{x-2} $$ using L'hopital
then $$\lim_{x\to2} \frac{e^{x^2-2x}-1}{x-2}=\lim_{x\to2} \frac{(2x-2)e^{x^2-2x}}{1}=2$$
Finally : $$\lim_{x\to2} \frac{e^{x^2}-e^{2x}}{(x-2)e^{2x}}=2$$