Can the limit of a quotient of trig functions approach a specific value?

In summary, the conversation discusses the limit of a quotient involving sin and cos as x approaches infinity. It is noted that sin and cos separately oscillate between [-1,1] and therefore their limits do not exist. The possibility of the quotient approaching a certain value is questioned, but the function must be rewritten in order to apply L'Hôpital's rule. However, the function cannot be rewritten to obtain a meaningful answer. Additionally, it is mentioned that the function is not defined where the denominator is zero and that there may be conditions that are not met for L'Hôpital's rule to be applied. The suggestion is made to carefully review the statement of L'Hôpital's rule and determine if any conditions are not
  • #1
Jonas
6
1
Homework Statement
Can lim x-> infinity sin(ln(x)) /cos(sqrt(x)) be evaluated with L'Hôpitals rule?
Relevant Equations
lim x→∞ sin(ln(x))/cos(√x)
Hello.
Sin and cos separately oscillates between [-1,1] so the limit of each as x approach infinity does not exist.
But can a quotient of the two acutally approach a certain value?
lim x→∞ sin(ln(x))/cos(√x) has to be rewritten if L'hôp. is to be applied but i can't seem to find a way to rewrite it to get a meaningful answer. Ofc i tried wolframalpha who states that it approaches + and - infinity which made me even more confused since i would normally interpret that as the limit does not exist?
 
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  • #2
The function isn't defined where the denominator is zero. And, in any case, if you analyse those points you'll see the problem.

It's simpler if you let ##y = \sqrt x## and take ##y \rightarrow \infty##.
 
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  • #3
You should look at the specific conditions where L'Hopital's rule can be used and see if those conditions are met. I suggest that you carefully review the statement of L'Hopital's rule. If you find one condition that is not met, state that condition and show that it is not met.
 
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Related to Can the limit of a quotient of trig functions approach a specific value?

1. Can the limit of a quotient of trig functions approach a specific value if the functions are undefined at that value?

Yes, it is possible for the limit of a quotient of trig functions to approach a specific value even if the functions are undefined at that value. This is because the limit is a concept that considers the behavior of the functions as they approach the specific value, rather than their behavior at the specific value itself.

2. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the functions as they approach the specific value from one side (either the left or the right), while a two-sided limit considers the behavior from both sides. The notation for a one-sided limit includes a plus or minus sign to indicate the direction of approach, while a two-sided limit is denoted with just an arrow pointing towards the specific value.

3. Can the limit of a quotient of trig functions approach a specific value if the functions are continuous at that value?

Yes, it is possible for the limit of a quotient of trig functions to approach a specific value even if the functions are continuous at that value. Continuity only guarantees that the functions have the same value at the specific value, but the limit considers the behavior of the functions as they approach the value.

4. How do you determine the limit of a quotient of trig functions?

To determine the limit of a quotient of trig functions, you can use algebraic manipulation and trigonometric identities to simplify the expression. Then, you can use the properties of limits to evaluate the limit. If the limit is still indeterminate, you can use L'Hopital's rule to find the limit.

5. Can the limit of a quotient of trig functions approach infinity?

Yes, it is possible for the limit of a quotient of trig functions to approach infinity. This can happen if the numerator of the quotient grows faster than the denominator as the input approaches a specific value. In this case, the limit is said to be divergent, as it does not approach a finite value.

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