Evaluate the following limit: _ x 0 a^x-a^ -x x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}-\text{a}^{\text{-x}}}{\text{x}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}-\text{a}^{\text{-x}}}{\text{x}}=\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{2x}-1}{\text{xa}^\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\big(\text{a}^{2\text{x}}-1\big)}{2\text{x}}\lim\limits_{\text{x}\rightarrow0}\frac{1}{\text{a}^\text{x}}$
$=2\text{log}_e 2$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(2 Marks)" from Limits→Limits — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

The area of a circle is $81$ TH sq.cm. Find the length of the arc subtending an angle of $300^\circ$ at the centre and also the area of corresponding sector.

Find the values of : tan 225°

Find the angle between hour-hand and minute-hand in a clock at : 10:10

Evaluate the following one sided limits:
$\lim\limits_{\text{x}\rightarrow2^+}\frac{\text{x}-3}{\text{x}^2-4}.$

Solve the following system of equations in R.
$\frac{2\text{x}+1}{7\text{x}-1}>5,\frac{\text{x}+7}{\text{x}-8}>2$

Use the binomial theorem to evaluate the following upto four places of decimals.

$(1.02)^{-5}$

From a pack of well-shuffled cards, two cards are drawn at random. Find the probability that both the cards are diamonds when

(i) the first card drawn is kept aside.

ii. the first card drawn is replaced in the pack.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{log(2+x)}+\text{log0.5}}{\text{x}}$

Find the sun of first n odd natural numbers.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{mx}-\text{b}^\text{nx}}{\sin\text{kx}}$