To evaluate the limit, we must first pull out a factor consisting of the highest degree term divided by itself (so we are unchanging the function):

After the factor we pulled out goes to 1, and all of the negative exponent terms go to zero (for they have infinity in their denominator, which equals zero), we are left with 

To evaluate the limit, we must first factor out a term consisting of the highest power term divided by itself (so we are unchanging the function inside the limit):

The term we factored out becomes 1, and all of the negative exponent terms as x goes to infinity go to zero (infinity in a denominator makes the whole term go to zero). We are left with

Step 1: Define what kind of function we have here..

We have a rational function..

Step 2: In general, if the value in the denominator is bigger than the numerator, the value of that fraction is getting close to 

Step 3: Plug in  into our function:





The limit of  as  is zero ()

Although this limit looks complex, we simply need to evaluate the function at the limit value.

To evaluate the limit as n goes to infinity, we must factor out a term consisting of the highest power term divided by itself (which is equal to one, so we are not technically changing the original function):

The factor we created goes to one, and the two terms in the denominator both equal zero as n goes to infinity (they are each terms with n in the denominator of a fraction, the denominator going to infinity, which equals zero), so the function itself has zero in the denominator, which equals .

To evaluate the limit, we must first see what side of the function we are evaluating the limit from. The plus sign exponent on 4 indicates we are approaching 4 from the right, using numbers slightly larger than 4. 

This corresponds to the second part of the piecewise function. When we evaluate the limit, . 

When we evaluate the limit using normal methods (substitution), we find that  is reached, which is an indeterminate form. To evaluate the limit, we can use L'Hopital's Rule, which states that when an indeterminate form is reached, 

Applying this to our limit, we get

The derivatives were found using the following rules:

, 

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 .