Evaluate the following limit: _ x 0 9^x-2.6^x+4^x x^2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{9^\text{x}-2.6^\text{x}+4^\text{x}}{\text{x}^2}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{9^\text{x}-2.6^\text{x}+4^\text{x}}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(3^\text{x}\big)^2-2.3^\text{x}2^\text{x}+\big(2^\text{x}\big)^2}{\text{x}^2} $
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{3^\text{x}-2^\text{x}}{\text{x}}\Big)^2$
$=\Big(\lim\limits_{\text{x}\rightarrow0}\frac{3^\text{x}-1}{\text{x}}-\lim\limits_{\text{x}\rightarrow0}\Big(\frac{2^\text{x}-1}{\text{x}}\Big)\Big)^2$
$=\Big(\text{log}\frac{3}{2}\Big)^2$

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 "2 Marks Questions" from Limits→Limits — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Which term of the sequence $\frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 }$, ... is $\frac { 1 } { 19683 }$?

Find the equation of a straight line:

with slope -2 and intersecting the x-axis at a distance of 3 units to the left of origin.

Prove that: $\begin{equation} 2 \sin ^{2} \frac{3 \pi}{4}+2 \cos ^{2} \frac{\pi}{4}+2 \sec ^{2} \frac{\pi}{3}=10 \end{equation}$

Prove that:
$
\sec \left(\frac{\pi}{4}+\theta\right) \sec \left(\frac{\pi}{4}-\theta\right)=2 \sec 2 \theta
$

How many words can be formed out of the letters of the word, 'ORIENTAL', so that the vowels always occupy the odd places?

Differentiate the following function with respect to $(\text{x})$:

$\frac{\text{x}^3}{3}-2\sqrt{\text{x}}+\frac{5}{\text{x}^2}$

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

If $A =\{1,2,6,9\}, B=\{2,4,6,8\}$ and $C=\{3,4,9\}$ then prove that $A \cup(B \cap C)=(A$ $\cup$ B) $\cap(A \cup C)$.

Prove that $\begin{equation} \frac{\cos 7 x+\cos 5 x}{\sin 7 x-\sin 5 x}=\cot x \end{equation}$

For all sets A, B and C, if $\text{A}\subset\text{B},$ then $\text{A}\cup\text{C}\subset\text{B}\cup\text{C}$