Given that limx→af(x)=0,limx→ag(x)=0,limx→ah(x)=1,limx→ap(x)=∞,limx→aq(x)=∞ which of the following limits are indeterminate forms? Fro those that are not an indeterminate form, evaluate the limit possible. limx→a[p(x)−q(x)]

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SoosteethicU · Accepted Answer

In this case, we consider limx→ap(x)=∞   limx→aq(x)=∞ and need to find the limit of the difference p(x)−q(x) as x→a. The best way to show that it is an indeterminate form is through examples. For example, let p(x)=2+x,q(x)=x and consider the following limits: limx→∞p(x)=limx→∞(2+x)=∞   limx→∞q(x)=limx→∞x=∞ Now, consider the limit as x→ ∞ of p(x)−q(x): limx→∞[p(x)−q(x)]=limx→∞[2+x−x]=limx→∞2=2 Therefore, we found an example in which the difference of two limits that approach infinity is finite. Now, let p1(x)=2+x,p2(x)=−x limx→∞p1(x)=limx→∞(2+x)=∞   limx→∞q1(x)=limx→∞(−x)=−∞ Now, consider the limit as x→∞ of p1(x)−q1(x): limx→∞[p1(x)−q1(x)]=limx→∞[2+x−(−x)]=limx→∞(2+x)=∞ Therefore, the difference of two limits that approach infinity can also be ∞ . This way, we can see that it is an indeterminate form ∞−∞, the limit depends on the given functions. Results: Indeterminate form.

Given that \lim_{x\rightarrow a}f(x)=0, \lim_{x\rightarrow a}g(x)=0, \lim_{x\

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