To evaluate the limit, we first must determine what side the limit is being evaluated on. The negative sign exponent on 2 indicates we are approaching 2 from the left side, or using numbers smaller than 2. Therefore, we must use the first half of the piecewise function. When evaluating the limit using the function corresponding to numbers less than or equal to 2, we get  (natural logarithm goes to negative infinity as the input goes to zero). 

To evaluate the limit, we can use the properties of exponents and factorials to make the following simplifications:

As n goes to infinity, the numerator goes to infinity which easily dominates the other constant terms.

When evaluating the limit, we must compare the terms in the numerator and denominator. In the numerator we have a polynomial term, while in the denominator we have an exponential term and a polynomial term (of a lesser degree than the numerator). Despite the polynomial in the numerator having a higher degree than the denominator, the exponential term grows faster, which means as n goes to infinity, the denominator goes to infinity faster. Infinity in the denominator equals .

To evaluate the limit, we can use properties of exponents and factorials to simplify:

The denominator goes to infinity as n goes to infinity, which makes the whole term go to zero.

When evaluating the limit, we must consider the terms in the limit. While the numerator and denominator both have a polynomial term of the same degree (which would suggest dividing their coefficients to get the limit), there is also an exponential term in the denominator, which grows faster than the polynomial terms. The denominator reaches infinity faster, which means the limit equals  (the overall term has a far larger term in the denominator than the numerator).

To evaluate the limit, we must factor out a term consisting of the highest power term divided by itself (which equals one, so we aren't changing the original function):

The term we factored goes to one, and the two terms with negative exponents in the denominator go to zero (they are each "fractions" with n in their denominator - the terms go to zero as the denominator goes to infinity), so we are left with .

To evaluate the limit, we can simplify the function using properties of factorials and exponents:

The simplified function has n only in the numerator; the function goes to infinity as n goes to infinity.

To evaluate the limit, we must first check to see from which side we are evaluating the limit. The negative sign exponent indicates that we are evaluating the limit from the left side, or using numbers smaller than 2. This corresponds to the first part of the piecewise function. When we evaluate the limit using this function, we get .

To evaluate the limit, we must compare the terms in the numerator and denominator. The numerator and denominator both have polynomial terms of the same degree. However, the numerator also has an exponential term that grows far faster than the polynomial terms. So, its growth dominates and the numerator goes to infinity, thus the limit of the overall function is .

To evaluate the limit, we must use the properties of exponents and factorials to simplify the function:

As n goes to infinity, the denominator of the simplified function gets infinitely large, which means the term goes to zero.