Aspire Faculty ID #11624 · Topic: NIMCET 2024 · Just now
NIMCET 2024

The value of the limit $$\lim _{{x}\rightarrow0}\Bigg{(}\frac{{1}^x+{2}^x+{3}^x+{4}^x}{4}{\Bigg{)}}^{1/x}$$ is

Solution

We need to find: $\displaystyle \lim_{x\to 0}\left( \frac{1^x + 2^x + 3^x + 4^x}{4} \right)^{1/x}$ 
Rewrite each term: $k^x = e^{x\ln k}$ 
So: $1^x + 2^x + 3^x + 4^x = e^{0} + e^{x\ln 2} + e^{x\ln 3} + e^{x\ln 4}$ 
As $x \to 0$ use expansion: $e^{x\ln k} = 1 + x\ln k + O(x^2)$ 
So: $1^x + 2^x + 3^x + 4^x$ 
$= 1 + (1 + x\ln 2) + (1 + x\ln 3) + (1 + x\ln 4)$ $= 4 + x(\ln 2 + \ln 3 + \ln 4) + O(x^2)$ 
Thus: $\frac{1^x + 2^x + 3^x + 4^x}{4} = 1 + \frac{x(\ln 2 + \ln 3 + \ln 4)}{4} + O(x^2)$ 
Now apply the standard limit: 
$\lim_{x\to 0} (1 + kx)^{1/x} = e^{k}$ 
Here: $k = \frac{\ln 2 + \ln 3 + \ln 4}{4}$ 
Hence the limit is: $e^{(\ln 2 + \ln 3 + \ln 4)/4}$ 
Combine logs: $\ln 2 + \ln 3 + \ln 4 = \ln(2\cdot 3 \cdot 4) = \ln 24$ 
Final answer: $\boxed{24^{1/4}}$
← Previous Next →

Previous 10 Questions — NIMCET 2024

Nearest first
1
The number of one - one functions f: {1,2,3} → {a,b,c,d,e} is
Topic: NIMCET 2024
 2
If (4, 3) and (12, 5) are the two foci of an ellipse passing through the origin, then the eccentricity of the ellipse is
Topic: NIMCET 2024

Next 10 Questions — NIMCET 2024

Ascending by ID
1
The value of m for which volume of the parallelepiped is 4 cubic units whose three edges are represented by a = mi + j …
Topic: NIMCET 2024
 2
The number of distinct real values of $\lambda$ for which the vectors ${\lambda}^2\hat{i}+\hat{j}+\hat{k},\, \hat{i}+{\…
Topic: NIMCET 2024
 3
There are 9 bottle labelled 1, 2, 3, ... , 9 and 9 boxes labelled 1, 2, 3,....9. The number of ways one can put these b…
Topic: NIMCET 2024
 4
If the perpendicular bisector of the line segment joining p(1,4) and q(k,3) has yintercept -4, then the possible values…
Topic: NIMCET 2024
 5
Let C denote the set of all tuples (x,y) which satisfy $x^2 -2^y=0$ where x and y are natural numbers. What is the card…
Topic: NIMCET 2024
 6
If $x=1+\sqrt[{6}]{2}+\sqrt[{6}]{4}+\sqrt[{6}]{8}+\sqrt[{6}]{16}+\sqrt[{6}]{32}$ then ${\Bigg{(}1+\frac{1}{x}\Bigg…
Topic: NIMCET 2024
 7
The number of solutions of ${5}^{1+|\sin x|+|\sin x{|}^2+\ldots}=25$ for $x\in(-\mathrm{\pi},\mathrm{\pi})$ is
Topic: NIMCET 2024
 8
The system of equations $x+2y+2z=5$, $x+2y+3z=6$, $x+2y+\lambda z=\mu$ has infinitely many solutions if 
Topic: NIMCET 2024
 9
Which of the following is TRUE?A. If $f$ is continuous on $[a,b]$, then $\int ^b_axf(x)\mathrm{d}x=x\int ^b_af(x)\mathr…
Topic: NIMCET 2024
 10
If F|= 40N (Newtons), |D| = 3m, and $\theta={60^{\circ}}$, then the work done by F actingfrom P to Q is
Topic: NIMCET 2024
Ask Your Question or Put Your Review.

loading...