Can we ignore $x^n$ against $e^x$ for any value of $n$ when $x$ is approaching to infinity.
someone told me that as $x$ approach to infinity value of $e^x$ is increasing towards infinity rapidly more than what $x^n$. so we can neglect $x^n$ in comparison to $e^x$ when both are in summation form like: $x^n + e^x +7 \approx e^x$
Since by Taylor series of exponential function is $$\ e^x = \sum \frac{x^n}{n!}.$$ Therefore $$ e^x > \frac{x^{n +1}}{(n+1)!}.$$ $$ \frac{x^n}{e^x} < \frac{(n+1)!}{x}. $$ Taking limits as $x\to\infty$, $$ \lim \frac{(n+1)!}{x} = 0 = \lim \frac{x^n}{e^x}$$ So yes we can