Lim x - 0tan mx -tan x

Lim_x→ 0tan mx/tan nx =lim_x→ 0tan mx/lim_x→ 0tan nx =lim_mx→ 0tan mx/mx× mx/lim_nx→ 0tan nx/nx× nx [ ∵ If x→ 0 then mx→ also nx→ 0 ] =1 × m/1 × n ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 0tan ...

Question

${lim}_{x \to 0} \frac{t a n m x}{t a n n x}$

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Solution

${lim}_{x \to 0} \frac{t a n m x}{t a n n x}$

$= \frac{{lim}_{x \to 0} t a n m x}{l i m_{x \to 0} t a n n x}$

$= \frac{{lim}_{m x \to 0} \frac{t a n m x}{m x} \times m x}{{lim}_{n x \to 0} \frac{t a n n x}{n x} \times n x}$

$[∵ I f x \to 0 t h e n m x \to a l s o n x \to 0]$

$= \frac{1 \times m}{1 \times n} [∵ {lim}_{x \to 0} \frac{t a n x}{x} = 1]$

$= \frac{m}{n}$

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limx→0tan mxtan nx

limx→0tan mxtan nx =limx→0tan mxlimx→0tan nx =limmx→0tan mxmx×mxlimnx→0tan nxnx×nx [∵If x→0 then mx→also nx→0] =1×m1×n [∵limx→0tanxx=1] =mn

${lim}_{x \to 0} \frac{t a n m x}{t a n n x}$