Factor the numerator.
A limit exists at a hole. What is the y-value of the hole?
Notice that this is a limit x→∞. Compare the greatest powers of the numerator and denominator.
This is another limit x→∞. Compare the terms with the highest powers in the numerator and denominator.
Think about the graph of y = sinx. It oscillates as x→∞. This means that it does not approaches a finite value AND does not approach ±∞. When a function oscillates, we say the limit does not exist.
Consider the numerator and the denominator separately.
As x→∞, y = sinx oscillates between y = −1 and y = 1.
As x→∞, y = x approaches ∞.
So this ratio has small values on the top and infinity on the bottom.