snipez90 said:Actually in this case it did not matter, and L'Hopital's rule could be applied again. However, if the limit can be computed without L'Hopital's rule, it's sometimes better to do without L'Hopital since you may overlook the hypotheses and try to apply it again incorrectly (this is probably not a big problem on second thought since most people have the 0/0, inf/inf cases ingrained in their heads).
The limit of ln(x^2+4) as x approaches infinity is infinity. This can be seen by the fact that as x gets larger, ln(x^2+4) also gets larger, approaching infinity.
The limit of x-5 as x approaches infinity is also infinity. This can be seen by the fact that as x gets larger, x-5 also gets larger, approaching infinity.
Both functions grow at the same rate as x approaches infinity, since their limits are both infinity. Therefore, they can be considered equivalent in terms of growth rate.
One way to prove this is to use the L'Hopital's rule, which states that if the limit of a function as x approaches infinity is infinity, then the limit of its derivative as x approaches infinity is also infinity. In this case, both ln(x^2+4) and x-5 have a derivative of 2x, which also approaches infinity as x approaches infinity.
Yes, a graph of both functions shows that they have the same growth rate as x approaches infinity. Both functions have an asymptote at y=infinity, indicating that their values increase without bound as x gets larger. Additionally, the two curves have similar shapes, further demonstrating their equivalent growth rates.