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  • Find the limit <img alt="\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}=" src="https://learn.careers360.com/latex-image/?%5Clim%20_%7Bx%20%5Crightarrow%200%7D%20%5Cfrac%7B1-%5Ccos%20m%2

Find the limit \lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}=

Option: 1

{m / n}


Option: 2

{n / m}


Option: 3

{\frac{m^{2}}{n^{2}}}


Option: 4

{\frac{n^{2}}{m^{2}}}


Answers (1)

best_answer

 

 

Limit Using Expansion (Part 2) -

Limit Using Expansion (Part 2)

\\\text { (vi) } \quad \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots \ldots .\\\\\text { (vii) } \quad \cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\ldots \ldots .\\\\\text { (viii) } \quad \tan x=x+\frac{x^{3}}{3}+\frac{2}{15} x^{5}+\ldots \ldots .\\\\\text { (ix) } \quad \sin ^{-1} x=x+\frac{1^{2} }{3 !}\cdot x^{3}+\frac{1^{2} \cdot 3^{2} }{5 !}\cdot x^{5}+\frac{1^{2} \cdot 3^{2} \cdot 5^{2} }{7 !} \cdot x^{7}\ldots \ldots \ldots\\\\\text { (x) } \quad \tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}+\ldots \ldots \ldots

 

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\text{As we know }\\ \cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\ldots \cdots \\ \lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}= \lim _{x \rightarrow 0} \frac{\frac{(mx)^{2}}{2 !}-\frac{(mx)^{4}}{4 !} \cdots}{\frac{(nx)^{2}}{2 !}-\frac{(mx)^{4}}{4 !} \cdots}=\frac{m^2}{n^2}

 

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