In this video I show by using l'Hospital's Rule for indeterminate limits, that Planck's Law for blackbody radiation approaches zero as the wavelength approaches zero or infinity. Trying to take the limits directly obtains an indeterminate result (either 0/0 or ∞/ ∞), which means we can apply l'Hospital's Rule and use the derivative of the numerator and denominator for the limit. In the case for the limit as wavelength approaches 0, we have to apply l'Hospital's Rule multiple time since several iterations of l'Hospital's Rule still obtain an indeterminate result of the form ∞/∞. Fortunately, a pattern arises in that each iteration obtains the same result as the previous one but with one less lambda (λ) or wavelength term. After 4 iterations of the l'Hospital's Rule, we find that the limit is in fact zero, thus also solving the ultraviolet catastrophe arising from the previous Rayleigh-Jeans Law which approached infinity at small wavelengths.
Timestamps:
- Question 1: Limits of Planck's Law using l'Hospital's Rule: 0:00
- Solution: Recap on l'Hospital's Rule for indeterminate limits: 0:28
- Grouping constants together in Planck's Law: 2:23
- Limit as wavelength approaches zero is indeterminate (∞/∞): 4:51
- Limit as wavelength approaches infinity is indeterminate (0/0): 6:36
- Applying l'Hospital's Rule for the limit at infinity: 7:55
- Limit of Planck's Law as wavelength approaches infinity is equal to zero: 15:02
- Applying l'Hospital's Rule for the limit at zero is still indeterminate (infinity / infinity): 15:33
- Applying 2nd iteration of l'Hospital's Rule for limit at zero is still indeterminate (infinity / infinity): 17:00
- Pattern shows that applying l'Hospital's Rule 3 more times the limit becomes zero: 21:23
- Limit of Planck's Law as wavelength approaches zero is equal to zero: 24:20
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