L'Hopital's Rule: Evaluating Limits

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SUMMARY

The forum discussion focuses on using l'Hopital's Rule to evaluate the limit lim x→0 e^(-1/x) / x for x > 0. Participants clarify that l'Hopital's Rule applies to limits of indeterminate forms such as 0/0 or ∞/∞. The limit in question does not initially appear to fit these criteria, but further analysis reveals that e^(-1/x) approaches 0 as x approaches 0 from the positive side, allowing for the application of the rule after appropriate differentiation.

PREREQUISITES
  • Understanding of l'Hopital's Rule and its application to limits
  • Knowledge of exponential functions and their behavior as x approaches 0
  • Familiarity with the concept of indeterminate forms in calculus
  • Basic differentiation techniques for functions
NEXT STEPS
  • Study the application of l'Hopital's Rule in various limit scenarios
  • Learn about exponential decay and its implications in calculus
  • Explore other methods for evaluating limits, such as algebraic manipulation
  • Investigate the concept of indeterminate forms in greater detail
USEFUL FOR

Students studying calculus, particularly those learning about limits and l'Hopital's Rule, as well as educators seeking to clarify these concepts for their students.

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Homework Statement



Use l'Hopital's rule to evaluate the following limit:
lim x→0 e^(-1/x) / x for x> 0.

Homework Equations



differentiate the top and bottom until a limit can be found. Possibly rewrite as a product.

The Attempt at a Solution


I was under the impression that l'Hopital's rule could only be used for evaluating limits of indeterminate form i.e. 0/0 or ∞/∞. The above quotient doesn't fall into this category, does it? If someone could clear this up for me it'd be great.
 
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DAPOS said:

Homework Statement



Use l'Hopital's rule to evaluate the following limit:
lim x→0 e^(-1/x) / x for x> 0.

Homework Equations



differentiate the top and bottom until a limit can be found. Possibly rewrite as a product.

The Attempt at a Solution


I was under the impression that l'Hopital's rule could only be used for evaluating limits of indeterminate form i.e. 0/0 or ∞/∞. The above quotient doesn't fall into this category, does it? If someone could clear this up for me it'd be great.
What is ##\lim_{x \to 0^+} e^{-1/x}##?
 
jbunniii said:
What is ##\lim_{x \to 0^+} e^{-1/x}##?

It approaches some very small number. Ok, I see now. Thanks!
 

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