Derivative using l'hopital's rule

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SUMMARY

The discussion centers on finding the derivative of the function f(x) defined as exp(-1/x) for x > 0 and 0 for x ≤ 0 using L'Hôpital's Rule. The user struggles to derive f'(0) and seeks clarity on applying L'Hôpital's Rule repeatedly without reaching a definitive value. It is established that f'(0) equals 0, but the user is unable to prove this through limits. The limit as x approaches 0 of exp(-1/x)/x leads to the limit of -exp(-1/x)/x when applying L'Hôpital's Rule.

PREREQUISITES
  • Understanding of L'Hôpital's Rule
  • Familiarity with limits in calculus
  • Knowledge of exponential functions
  • Basic concepts of derivatives
NEXT STEPS
  • Study the application of L'Hôpital's Rule in more complex limits
  • Explore the definition of the derivative in detail
  • Investigate the behavior of exponential functions near zero
  • Learn about continuity and differentiability at points of discontinuity
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Students and educators in calculus, mathematicians focusing on limits and derivatives, and anyone seeking to understand the application of L'Hôpital's Rule in complex scenarios.

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let f(x)

exp(-1/x) for x>0, 0 for x<=0

i want to get f'(x) by using l'hopital's rule, but somehow

i'm applying l'hopital's rule again and again and no clear value is coming out.

i know f'(0) is 0, but i cannot prove it
 
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So, using the definition of the derivative, you're looking for the limit as x-->0 of exp(-1/x)/x.

If you use L'Hopital on this, you get that it is equal to the limit as x-->0 of -exp(-1/x)/x.

So...?
 

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