# A discussion on the two limit cases of sin

## Limits calculus

This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space. And for those of you who say, hey, we already knew how to do this. Now this thing right over here is pretty straightforward. It just means that we moved the wrong function to the denominator. The natural log of Y is equal to, if we just rewrite this this is going to be the natural log of sine of X, the natural log of sine of X over the natural log of X. Also, if the denominator is going to infinity, in the limit, we tend to think of the fraction as going to zero. Well to get the natural log of something that gets you one, well Y must be approaching E because natural log of E is one. To look a little more into this, check out the Types of Infinity section in the Extras chapter at the end of this document. There's justifications why this could be zero, justifications why this could be one.

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.

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