Lim x - 0 a mx - b nx -sin kx

Lim_x→ 0a^mx b^nx/sin kx =lim_x→ 0(a^mx 1) (b^nx 1)/sin kx =lim_x→ 0a^mx 1/x b^nx 1/x/sin kx/x =lim_x→ 0a^mx 1/mx× m lim_x→ 0b^nx 1/nx× n/lim_x→ 0sin kx/kx× k = ...

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Byju's Answer

Standard XII

Mathematics

Sandwich Theorem

lim x → 0 a m...

Question

${lim}_{x \to 0} \frac{a^{m x} - b^{n x}}{s i n k x}$

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Solution

${lim}_{x \to 0} \frac{a^{m x} - b^{n x}}{s i n k x}$

$= {lim}_{x \to 0} \frac{(a^{m x} - 1) - (b^{n x} - 1)}{s i n k x}$

$= {lim}_{x \to 0} \frac{\frac{a^{m x} - 1}{x} - \frac{b^{n x} - 1}{x}}{\frac{s i n k x}{x}}$

$= \frac{{lim}_{x \to 0} \frac{a^{m x} - 1}{m x} \times m - {lim}_{x \to 0} \frac{b^{n x} - 1}{n x} \times n}{{lim}_{x \to 0} \frac{s i n k x}{k x} \times k}$

$= \frac{m l o g a - n l o g b}{k} = \frac{l o g a^{m} - l o g b^{n}}{k}$

$= \frac{1}{k} l o g (\frac{a^{m}}{b^{n}})$

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${lim}_{x \to 0} \frac{a^{m x} - b^{n x}}{x}$

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$Let f (x) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a (x) & b (x) & c (x) m (x) & n (x) & l (x) g (x) & h (x) & k (x) \end{matrix} ∣ ∣ ∣ ∣ ∣ then$

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limx→0amx−bnxsin kx

limx→0amx−bnxsin kx =limx→0(amx−1)−(bnx−1)sin kx =limx→0amx−1x−bnx−1xsin kxx =limx→0amx−1mx×m−limx→0bnx−1nx×nlimx→0sin kxkx×k =mloga−nlogbk=logam−logbnk =1klog(ambn)

${lim}_{x \to 0} \frac{a^{m x} - b^{n x}}{s i n k x}$