## Other Indeterminate Quotient Forms We have discussed the indeterminate quotient form . But there some indeterminate quotient forms similar to this one. Indeed, since then one can see that the indeterminate quotient forms as well as the indeterminate forms may be easily converted to the indeterminate quotient form . So first notice that Hôpital's rule is still valid when dealing with the indeterminate quotient form . Also worth to mention that the point a may be finite or infinite, Hôpital's rule still applies.

Example. Find the limit Answer. Fix . We have Hence We will use Hôpital's rule. We have So clearly we will keep to use Hôpital's rule n-times to get to the function Since Therefore, we have Note that this limit implies that, though both functions are very large when , the exponential function is more powerful than the power function (in fact more powerful than any polynomial function). We will write Example. Find the limit Answer. Fix . We have Hence We will use Hôpital's rule. We have Hence we have which implies Clearly this example implies that Putting the two examples together we conclude that when .

The indeterminate forms The main idea behind these indeterminate forms is to transform 0 into (the depends on whether we have 0+ or 0-), or transform into . This will lead to the indeterminate quotient forms Practically, you will be given a product f(x)g(x) where one function goes to 0 while the other one goes to . So you will use the following algebraic manipulations Example. Find the limit  Hence Rewrite the given expression into Computing the limit we will find Here we have a choice. We may use Hôpital's rule or Taylor Polynomials. In any case, Hôpital's rule is not bad to use in this case. Indeed, we have Since we conclude that Note that when , we have and which imply May be this is easier, what do you think???

Example. Find the limit  We will ask you to check that whether you take or The calculations are not easy. Here let us show how some tricks may help. First switch from x into t = 1/x. We will have Note that when . Next, we use (when ), to get  Therefore, we have  [Geometry] [Algebra] [Differential Equations]
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