Lim x - 05 x o s x-3 sin x-3 x2-tan x

Lim_x → 05xcosx+3sinx/3x^2+tanx Dividing numerator and denominator by x we get lim_x → 05cosx+3sinx/x/3x+tanx/x =lim_x → 05cosx+lim_x → 03sinx/x/lim_x → 0 3x+l ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 05 x ...

Question

${lim}_{x \to 0} \frac{5 x c o s x + 3 s i n x}{3 x^{2} + t a n x}$

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Solution

${lim}_{x \to 0} \frac{5 x c o s x + 3 s i n x}{3 x^{2} + t a n x}$

Dividing numerator and denominator by x we get

${lim}_{x \to 0} \frac{5 c o s x + 3 \frac{s i n x}{x}}{3 x + \frac{t a n x}{x}}$

$= \frac{{lim}_{x \to 0} 5 c o s x + {lim}_{x \to 0} \frac{3 s i n x}{x}}{{lim}_{x \to 0} 3 x + {lim}_{x \to 0} \frac{t a n x}{x}}$

$= \frac{5 {lim}_{x \to 0} c o s x + 3 {lim}_{x \to 0} \frac{s i n x}{x}}{3 {lim}_{x \to 0} x + {lim}_{x \to 0} \frac{t a n x}{x}}$

$= \frac{5 \times c o s 0 + 3 \times 1}{3 \times 0 + 1}$

$[∵ {lim}_{x \to 0} \frac{s i n x}{x} = 1 and {lim}_{x \to 0} \frac{t a n x}{x} = 1]$

=8

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limx→05xcosx+3sinx3x2+tanx

limx→05xcosx+3sinx3x2+tanx Dividing numerator and denominator by x we get limx→05cosx+3sinxx3x+tanxx =limx→05cosx+limx→03sinxxlimx→0 3x+limx→0tanxx =5 limx→0cosx+3 limx→0sinxx3 limx→0x+limx→0tanxx =5×cos0+3×13×0+1 [∵limx→0sinxx=1 andlimx→0tanxx=1] =8

${lim}_{x \to 0} \frac{5 x c o s x + 3 s i n x}{3 x^{2} + t a n x}$