Evaluate the limit as x goes to infinity using L'Hospital's rule

In summary, the conversation is discussing how to use L'Hospital's Rule to evaluate the limit as x goes to infinity for the expression 8xe^(1/x)-8x. The person asks for clarification on how to approach the problem when the denominator is 1, and the conversation ends with a hint to factorize and use the fact that x is equal to 1/(1/x).
  • #1
Loopas
55
0
(1)

Evaluate the limit as x goes to infinity using L'Hospital's rule:

8xe^(1/x)-8x

(2)

L'Hopital's Rule

(3)

How can I use L'Hopital's Rule for this problem if the denominator is 1? Wouldn't that just give me an undefined limit? This may be a pretty stupid question, but I'm new to this.
 
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  • #2
Loopas said:
(1)

Evaluate the limit as x goes to infinity using L'Hospital's rule:

8xe^(1/x)-8x

(2)

L'Hopital's Rule

(3)

How can I use L'Hopital's Rule for this problem if the denominator is 1? Wouldn't that just give me an undefined limit? This may be a pretty stupid question, but I'm new to this.

Hint: factorise first. Also, ##x = \displaystyle \frac{1}{\frac{1}{x}}##.
 
  • #3
How do you you know that x=(1)/(1/x)?
 
  • #4
[tex]\frac{1}{(\frac{1}{x})}[/tex]

is the reciprocal of 1/x...

ehild
 
  • #5
Ahhh ok so I can rewrite as (8x(e^(1/x)-1))/(1/x)?
 
  • #6
Not quite.
 
  • #7
Loopas said:
Ahhh ok so I can rewrite as (8x(e^(1/x)-1))/(1/x)?

That x is replaced by the 1/(1/x) term. So why does it appear again?

Sorry no latex. On my phone at the moment.
 

1. What is L'Hospital's rule?

L'Hospital's rule is a mathematical principle used to evaluate limits involving indeterminate forms, such as 0/0 or infinity/infinity. It states that if the limit of a function f(x) divided by g(x) as x approaches a specific value is an indeterminate form, then the limit of the quotient of their derivatives is equal to the original limit.

2. When should I use L'Hospital's rule?

L'Hospital's rule should be used when evaluating limits involving indeterminate forms. These types of limits are often encountered in calculus, and L'Hospital's rule provides a way to solve them without having to use more complicated techniques.

3. How do I use L'Hospital's rule?

To use L'Hospital's rule, first determine if the limit is an indeterminate form. If it is, take the derivative of the numerator and denominator separately. Then, evaluate the limit of the quotient of the derivatives. If the resulting limit is still an indeterminate form, repeat the process until a definitive answer is obtained.

4. Can L'Hospital's rule be used for limits at infinity?

Yes, L'Hospital's rule can be used to evaluate limits at infinity. In this case, the limit of the quotient of the derivatives will be equal to the limit of the original function as x approaches infinity.

5. Are there any limitations to using L'Hospital's rule?

Yes, there are some limitations to using L'Hospital's rule. It can only be used for limits involving indeterminate forms, and it may not always provide a definitive answer. Additionally, it is important to check the validity of using L'Hospital's rule, as it may not be applicable in all cases.

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