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If f(x) = sin(x), find
This limit is not defined. Look at the graph of f(x):
As x gets bigger and bigger, f will sometimes be 1 and sometimes -1. There is no one value that f(x) is approaching - it's bouncing around all over the place. This limit does not exist.
If f(x) = x + 1, find
As x gets bigger, f(x) keeps getting bigger, too. There is no number that f(x) is approaching, because f(x) keeps growing without bound. This limit does not exist. We could also say
since f(x) is forever getting larger and is unbounded.
If we have a rational function where the degree of the numerator is smaller than the degree of the denominator, what happens to the value of that rational function as x approaches ∞?
Use a table to estimate
What happens as x gets larger and larger?
It looks like the quotient is approaching 0, so we'll say
The same will be true for any rational function where the degree of the numerator is smaller than the degree of the denominator. We know
because we're dividing 1 by larger and larger things as x approaches infinity. If we have a rational function
where the degree of p(x) is smaller than the degree of q(x), q will get larger "faster" than p will, and the fraction will approach 0.
We'll talk more about this in a bit, and include more pictures, when we compare functions and their limits at infinity more generally.