To evaluate the limit, first pull out the highest power term out of the numerator and denominator (so essentially you are pulling 1):

As you can see, the  and  terms as they approach infinity go to zero. What is left over is .

To evaluate this limit easily, simply pull out a factor of the highest degree term over the highest degree term (1):

As you can see, after the  divides to 1, then the denominator becomes 1 and the numerator becomes 0.

The final answer is therefore .

To evaluate the limit, first pull out a term equivalent to 1 - in this case, the highest degree term:

Once the "1" term cancels, and the  term goes to zero (as you can see, the fraction  goes to zero as the denominator gets infinitely huge), all that is left is , our final answer.

To evaluate the limit, simply pull out a factor of the highest degree term divided by the highest degree term (so you are simply multiplying by 1):

After the highest degree terms cancel out, what you have left over are three terms in which the denominator of the fraction goes to infinity, making those three terms go to zero. What you have left then is our final answer, 

.

To evaluate this limit, pull out a fraction consisting of the highest power term over the highest power term (so you are pulling out 1):

After the term we pulled out cancels to become 1, it is easy now to see that the negative power terms all go to zero as the limit approaches infinity.

We have zero in the numerator of the fraction, so our answer is .

To evaluate the limit, pull out a factor of the highest degree term divided by the highest degree term (we are essentially removing a 1):

After the term we pulled out cancels to 1, it is easier to evaluate the limit.

All of the terms with negative powers go to zero as the limit approaches infinity, and that leaves zero in the numerator, meaning our answer is .

To evaluate the limit, pull out a factor of the highest degree term divided by the highest degree term (1):

After the term we pulled out cancels to 1, what we have left is a term that goes to zero (negative exponent means infinity in the denominator), and a constant  which remains at the end as our answer. 

To evaluate this limit easily, first pull out a term consisting of the highest degree term divided by the highest degree term (so we are essentially removing 1):

After the term we made cancels to one, you can see that all of the terms with negative exponents go to zero when the limit approaches infinity.

We are then left with 

.

The limit we are given for the piecewise function is not one-sided. Therefore, in order for the limit to exist, it must approach the same value from the right and left sides. From the left side, we approach , but from the right side, we approach , which are not equal, and therefore the limit does not exist.

For the limit given, we have a polynomial function on top, and on the bottom we have an exponential function added to a polynomial function. Despite the numerator having a polynomial function of a higher degree than the denominator, the exponential function grows far faster than the polynomial function, thus the limit equals zero as  approaches infinity.