1: When n is large, the expression 3 − n² is always negative, so the absolute value can be replaced with a negative sign and thus removed. Also, when n grows large, then n² becomes the dominant term in the numerator and 2ⁿ in the denominator, so the other terms can be ignored. And since "exponentials beat powers", the 2ⁿ in the denominator eventually prevails and the fraction tends to zero. Thus the answer to the limit should be e⁰ = 1. Note, however, that this guessing was not as reliable as it usually is, since we were ignoring parts in exponentials, which can be treacherous.

How can this be proved? To do the above reasoning formally, one should factor out the dominant terms

Then it only remains to prove that the fractions in the last step indeed go to zero, which can be done by the l'Hospital rule (the fractions are of the indeterminate type).

Alternative solution would use the fact that the original ratio (after getting rid of the absolute value) is also of the type infinity over infinity, so the l'Hospital rule can be applied directly to it. Of course, this rule cannot be applied to powers, only fractions, but fortunately exponential is a nice function and as such it can be pulled out of the limit. Then one should remember to put the result (limit of the ratio) back to the exponential.

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