Question

limit x tends 0 2 power x - 3power x /x

Answer 1

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Answer 2

Given:

$\lim_{x \to 0} \dfrac{2^{x}-3^{x}}{x}$

We have to find, the value of $\lim_{x \to 0} \dfrac{2^{x}-3^{x}}{x}$ is:

Solution:

∴ $\lim_{x \to 0} \dfrac{2^{x}-3^{x}}{x}$

Adding and subtracting 1 by numerator, we get

= $\lim_{x \to 0} \dfrac{(2^{x}-1)-(3^{x}-1)}{x}$

= $\lim_{x \to 0} \dfrac{(2^{x}-1)}{x}-\lim_{x \to 0} \dfrac{(3^{x}-1)}{x}$

Using the limit identity:

$\lim_{x \to 0} \dfrac{(a^{x}-1)}{x}$= $\log a$

= $\log 2$ - $\log 3$

Using the logarithm identity:

$\log \dfrac{a}{b}$ = $\log a$ - $\log b$

= $\log \dfrac{2}{3}$

∴ $\lim_{x \to 0} \dfrac{2^{x}-3^{x}}{x}$ = $\log \dfrac{2}{3}$

Thus, the value of $\lim_{x \to 0} \dfrac{2^{x}-3^{x}}{x}$ is $\log \dfrac{2}{3}$ .