So if the limit of these guys' derivatives exist-- or the rational function of the derivative of this guy divided by the derivative of that guy-- if that exists, then this limit's going to be equal to the limit as x approaches infinity of-- arbitrarily switch colors-- derivative of 8x minus 5 is just 8.
Then Y must be approaching E and we are done because that's what we cared about.
So you have an x squared times 4 minus 5 over x. And we'll do that in a second. Example 1 Example 1 Evaluate each of the following limits. Which one of these two we get after doing the rewrite will depend upon which fact we used to do the rewrite.
The former spelling is still used in English where there is no circumflex. We now know and I'll write it out in language. Her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system.
That's not in the domain for the natural log but as you get closer and closer to zero from the negative direction, the natural log of those values, you have to raise E to more larger and larger negative values. This might ring a bell.