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				   1.   If f(x),g(x),h(x) are continuous for all x belongs to 'R'. and
h(x)=Lt n→∞ f(x)+x2ng(x)/1+x2n , f(1)/g(1)=K, f(-1)/g(-1)=L then k+L=?
2. A and B are two fixed points on fixed circle and 'p' is a moving    point on the above circle,angleAPB=60 , AB=8, range of PA*PB is ?
3. The population of a country increases by 2% per year.In 100 years population  increases by 'p' times then [p]=?  ([.] denotes greatest integer function)



7 years ago

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										Ans:3.Population increases by 2% every year. Let ‘p’ be the population at any time & p0be the initial population.Given:$\frac{dp}{dt} = .02p$$\int \frac{dp}{p} = \int .02dt$$\int_{p_{0}}^{p} \frac{dp}{p} = \int_{0}^{100} .02dt$$(lnp)_{p_{0}}^{p} = (.02t)_{0}^{100}$$ln\frac{p}{p_{0}} = (.02.100) = 2$$p = p_{0}e^{2}$Increase in the population:$p - p_{0} = p_{0} e^{2} -p_{0} = p_{0} (e^{2}-1)$$[e^{2}-1] = [2.718^{2}-1] = [6.3875] = 6$3.What is the range. Its not clear. Please rewrite this part again.1.$h(x) = \lim_{n\rightarrow \infty }(f(x) + \frac{x^{2n}.g(x)}{1+x^{2n}})$Since functions h(x), f(x) & g(x) are continuous in R. There limit should exist at -1 & 1.Limit at x =1LHL = RHL$\lim_{x\rightarrow 1^{-}}h(x) = \lim_{x\rightarrow 1^{+}}h(x)$$LHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(1-h)+\frac{g(1-h).(1-h)^{2n}}{1+(1-h)^{2n}})$$LHL = f(1)$$RHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(1+h)+\frac{g(1+h).(1+h)^{2n}}{1+(1+h)^{2n}})$$RHL = f(1) + g(1)$$LHL =RHL$$f(1) = f(1)+g(1)$$\Rightarrow g(1) = 0$Limit at x = -1$LHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(-1-h)+\frac{g(-1-h).(-1-h)^{2n}}{1+(-1-h)^{2n}})$$LHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(-1-h)+\frac{g(-1-h).(1+h)^{2n}}{1+(1+h)^{2n}})$$LHL = f(-1) + g(-1)$$RHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(-1+h)+\frac{g(-1+h).(-1+h)^{2n}}{1+(-1+h)^{2n}})$$RHL = \lim_{n\rightarrow \infty }\lim_{h\rightarrow 0}(f(-1+h)+\frac{g(-1+h).(1-h)^{2n}}{1+(1-h)^{2n}})$$RHL = f(-1)$$LHL=RHL$$\Rightarrow g(-1) = 0$Thanks & RegardsJitender SinghIIT DelhiaskIITians Faculty

2 years ago

# Other Related Questions on Differential Calculus

Plz help me solve this question. It is from chapter limits and derivatives . I did not find that in the list,

Krishna, write your questions that you want to ask, to related to limits and derivative. But you studied the defnition of limits and derivative clearly.

 Kumar 3 months ago

Try desmos (Google it ) ... It is used to graph the function.. Then you can set y as a parameter and then you will see that the function as x-> 0 the graph tends to infinity (for some values...

 rishabh 3 months ago

Kumar the question is lim x--> 0 ((y^2 - (y-x)^2)^1/x). / (8xy - 4x^2 + (8xy)^1/2)^3 I have also attached the image of it with the question please take a look

 Sujith krishna 3 months ago
Why does lhl and rhl are not equal in this question.w hat should i do in these type of questions

coz this is not 0/0 inderminate form ...it is exact 0/0 which is not defined(n.d.)

 Nishant Vora one month ago

 Nishant Vora one month ago

Why cannot we write tan[x]/[x] as 1 when x tends towards zero..and if is in 0/0 form why didnt u used lhospital rule

 Samaksh one month ago
i can not solve this inequality please sir how to solve this inequality Ix+1I+Ix-2I

i am trying to write conditions in the answer but after posting the conditions are vanishing , you can see above, dont know why is it happening.... :(

 DR STRANGE 12 days ago

Ix+1I+Ix-2I = 2x-1 ….....for x>2 =x+1 -(x-2) = 3 …......for -1 = -(x+1) -(x-2) = 1-2x ….......for x

 DR STRANGE 12 days ago

mistake............................................................... =3 for...... -1 = 1-2x ….for x

 DR STRANGE 12 days ago
If alpha is a real root of the equation ax 2 +bx+c and beta is a real root of equation -ax 2 +bx+c. Show that there exists a root gama of the equation (a/2)x 2 +bx+c which lies between alpha...

 Ajay 6 months ago

Small Mistake in last para posting again..............................................................................................................

 Ajay 6 months ago

We have Similarly, So if P(x) = a/2 x 2 +bx +c, then and are off opposite sign and hence there must exist a root between the two numbers.

 mycroft holmes 6 months ago
In the listed image can you tell me how beta*gamma = 2 ….. . . .. ??

The value of gamma is still not correct, either printing mistake or you gave me wrong value. The correct value of gamma is below

 Ajay 5 months ago

Thankyou so much............................. …......................................................................!

 Anshuman Mohanty 5 months ago

Yes sorry..... . . . .it is not so clear.. ok the values are beta = α + α^2 + α^4 and gamma = α^3 + α^5 + α^7

 Anshuman Mohanty 5 months ago
if |z - i| Options: a*) 14 b) 2 c) 28 d) None of the above

If |z-i| = ?? PLs complete the question

 Nishant Vora one month ago

Got it! [z + 12 – 6 i ] can be rewritten as [ z – i + 12 – 5 i] => | z – i | and => |12 – 5 i | = sqrt ( 12^2 + 5^2) = 13......................(2) => | z + 12 – 6 i | => | z + 12 – 6 i |...

 Divya one month ago

I tried posting the question several times, it kept cutting off the rest of the question. Here: If | z-1| Options: a*) 14 b) 2 c) 28 d) None of the above

 Divya one month ago
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