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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits2 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}+\text{b}^\text{x}-\text{c}^\text{x}-\text{d}^\text{x}}{\text{x}}$
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}+\text{b}^\text{x}-\text{c}^\text{x}-\text{d}^\text{x}}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}-1}{\text{x}}+\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}-1}{\text{x}}-\lim\limits_{\text{x}\rightarrow0}\frac{\text{c}^\text{x}-1}{\text{x}}-\lim\limits_{\text{x}\rightarrow0}\frac{\text{d}^\text{x}-1}{\text{x}}$
$=\text{log a}+\text{log b}-\text{log c}-\text{log d}$
$=\text{log}\Big(\frac{\text{ab}}{\text{cd}}\Big)$

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