Problem: Evaluate the following limit (if it exists)

Solution: What do we get if we try to plug in infinity? 3^∞ gives infinity, but (−2)^∞ does not have a limit. This shows that we do not get the answer by limit algebra.

So we have to try something. Since we do not have a limit for some terms, this sequence is not of any popular type of operations (like indeterminate ratio). Is it at least some kind of expression for which we have a special "box"? Fortunately yes, this example fits the box "polynomials and ratios with powers". So we know that it should be solvable by factoring dominant terms. In order to identify them we have to first simplify the given fraction:

We see that 3ⁿ is the dominant term in the numerator and 4ⁿ is the dominant term in the denominator. We factor them out:

Note how in those little fractions we pulled out n as a common exponent. This is the usual way to handle fractions of the form aⁿ/bⁿ, turning them into geometric sequences.

Now we are ready to evaluate the limit. Note that we have lots of geometric sequences there. All of them have the property that their bases are less than one (in absolute value) and therefore they converge to zero:

Is there any other way? Not really. Note that we cannot pass to functions and try some trick from that part (like l'Hospital), because we cannot consider in general the power (−2)^x; indeed, we only have exponentials for positive bases.

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