Lim x - 0 a mx - b nx - x

Lim_x→ 0a^mx b^nx/x lim_x→ 0(a^mx 1) (b^nx 1)/x =lim_x→ 0ma^mx 1/mx lim_x→ 0nb^nx 1/nx =m log a n log b =log ( a^m/b^n ) ...

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Byju's Answer

Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

lim x → 0 a m...

Question

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

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Solution

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

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

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

$= m l o g a - n l o g b$

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

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Similar questions

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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−bnxx

limx→0amx−bnxx limx→0(amx−1)−(bnx−1)x =limx→0mamx−1mx−limx→0nbnx−1nx =m log a−n log b =log(ambn)

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

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

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

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

$= m l o g a - n l o g b$

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