Lim-x- 0a-x-b-x-c-x-d-x-x

Lim_x→ 0a^x+b^x c^x d^x/x =lim_x→ 0(a^x 1)+(b^x 1) (c^x 1)+(d^x 1)/x =lim_x→ 0a^x 1/x+lim_x→ 0b^x 1/x lim_x→ 0c^x 1/x lim_x→ 0d^x 1/x =log a+log b log c log d = ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim_x→ 0a^x+b...

Question

${lim}_{x \to 0} \frac{a^{x} + b^{x} - c^{x} - d^{x}}{x}$

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Solution

${lim}_{x \to 0} \frac{a^{x} + b^{x} - c^{x} - d^{x}}{x}$

$= {lim}_{x \to 0} \frac{(a^{x} - 1) + (b^{x} - 1) - (c^{x} - 1) + (d^{x} - 1)}{x}$

$= {lim}_{x \to 0} \frac{a^{x} - 1}{x} + {lim}_{x \to 0} \frac{b^{x} - 1}{x} - {lim}_{x \to 0} \frac{c^{x} - 1}{x} - {lim}_{x \to 0} \frac{d^{x} - 1}{x}$

$= l o g a + l o g b - l o g c - l o g d$

$= l o g (\frac{a b}{c d})$

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limx→0ax+bx−cx−dxx

limx→0ax+bx−cx−dxx =limx→0(ax−1)+(bx−1)−(cx−1)+(dx−1)x =limx→0ax−1x+limx→0bx−1x−limx→0cx−1x−limx→0dx−1x =log a+log b−log c−log d =log(abcd)

${lim}_{x \to 0} \frac{a^{x} + b^{x} - c^{x} - d^{x}}{x}$