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  4. Let l= underset x arrow 0+ textLim x m ( ln x ) n where m , n ∈ N then :

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Q. Let $l=\underset { x \rightarrow 0^{+}}{\text{Lim}} x ^{ m }(\ln x )^{ n }$ where $m , n \in N$ then :

Continuity and Differentiability

Solution:

$l=\underset { x \rightarrow 0^{+}}{\text{Lim}} \frac{(\ln x )^{ n }}{ x ^{- m }}$
Using L'Hospital's rule $l\underset { x \rightarrow 0^{+}}{\text{Lim}} \frac{ n !(-1)^{ n }}{ m ^{ n }} x ^{ m }=0$