Find the value of lim x - 0ax-1-x

The correct option is C We have,lim_x→ 0a^x 1/x It is an examle of 0/0 form While evaluating the limit,we can use expansion of a^x. We know, a^x=1+x(log_ea)+x ...

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Standard XII

Mathematics

Many-One onto Function

Find the valu...

Question

Find the value of $lim x \to 0 \frac{a^{x} - 1}{x}$

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Solution

The correct option is B

We have, $lim x \to 0 \frac{a^{x} - 1}{x}$

It is an examle of $\frac{0}{0}$ form

While evaluating the limit,we can use expansion of $a^{x}$ .

We know,

$a^{x} = 1 + x (l o g_{e} a) + \frac{x^{2}}{2!} (l o g_{e} a)^{2} + . . . . . .$

substituting the expansion of $a^{x}$ in the numerator of the limit

$= lim x \to 0 \frac{[1 + x (l o g_{e} a) + \frac{x^{2}}{2!} (l o g_{e} a)^{2} + . . . . . . .] - 1}{x}$

$= lim x \to 0 l o g_{e} a + \frac{x}{2!} (l o g_{e} a)^{2} + . . . . . .$

$= l o g_{e} a + 0$

$= l o g_{e} a$

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Similar questions

Q. Use the formula

lim x \to 0 \frac{a^{x} - 1}{x} = log a

to find

lim x \to 0 \frac{2^{x} - 1}{{(1 + x)}^{1 / 2} - 1}

which is

= a log a

. Find

a

lim x \to 0 \frac{a^{x} - b^{x}}{x}

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

lim x \to 0 \frac{a^{x} - 1}{b^{x} - 1}

\lim_{x \to 0} \frac{a^{x} + a^{- x} - 2}{x^{2}}

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Find the value of limx→ 0ax−1x

Find the value of $lim x \to 0 \frac{a^{x} - 1}{x}$