Home
Class 12
MATHS
If x(1) and x(2) are the real and distin...

If `x_(1)` and `x_(2)` are the real and distinct roots of `ax^(2)+bx+c=0` then prove that `lim_(xtox1) (1+sin(ax^(2)+bx+c))^((1)/(x-x_(1)))=e^(a(x_(1)-x_(2))).`

A

does not exist

B

1

C

`oo`

D

`(1)/(2)`

Text Solution

Verified by Experts

`ax^(2)+bx+c=a(x-x_(1))(x-x_(2))`
`underset(xtox1)lim(1+sin(ax^(2)+bx+c))^((1)/(x-x_(1)))" "(1^(oo))" form")`
`=e^(underset(xtox_(1))lim(sin(a(x-x_(1))(x-x_(2))))/((x-x_(1))))`
`=e^(underset(xtox_(1))lim(sin(a(x-x_(1)).(x-x_(2))))/(a(x-x_(1))(x-x_(2))).a(x-x_(2)))`
`=e^(a(x_(1)-x_(2)))`
Promotional Banner

Topper's Solved these Questions

  • LIMITS

    CENGAGE ENGLISH|Exercise EXERCISE 2.8|8 Videos

  • LIMITS

    CENGAGE ENGLISH|Exercise Exercises (Single Correct Answer Type)|76 Videos

  • LIMITS

    CENGAGE ENGLISH|Exercise EXERCISE 2.6|9 Videos

  • JEE 2019

    CENGAGE ENGLISH|Exercise Chapter 10|9 Videos

  • LINEAR COMBINATION OF VECTORS, DEPENDENT AND INDEPENDENT VECTORS

    CENGAGE ENGLISH|Exercise DPP 1.2|10 Videos

Similar Questions

Explore conceptually related problems

If x_1 and x_2 are the real and distinct roots of a x^2+b x+c=0, then prove that lim_(n->x_1){1+"sin"(a x^2+b x+c)}^(1/(x-x_1))=e^(a(x_1-x_2))

If alpha an beta are roots of ax^(2)+bx+c=0 the value for lim_(xto alpha)(1+ax^(2)+bx+c)^(2//x-alpha) is

Let alpha and beta be the distinct roots of ax^(2) + bx + c = 0 . Then underset(x to alpha)(lim) (1 - cos (ax^(2) + bx + c))/((x - alpha)^(2)) equal to

Prove that Lim_(x to 0) (root3((1+x))-root3((1-x)))/x = 2/3

If 1,2,3 are the roots of the equation x^(3) + ax^(2) + bx + c=0 , then

If alpha and beta are roots of the equation ax^2+bx +c=0 , then lim_(xrarralpha) (1+ax^2+bx+c)^(1//x-alpha) , is

If alpha is a repeated root of ax^2+bx +c=0 , then lim_(xrarralpha)(sin(ax^2+bc+c))/(x-alpha)^2 is equal to

If x_(1) and x_(2) are the arithmetic and harmonic means of the roots fo the equation ax^(2)+bx+c=0 , the quadratic equation whose roots are x_(1) and x_(2) is

If L=lim_(xto0) (1)/(x^(3))((1)/(sqrt(1+x))-(1+ax)/(1+bx)) exists,then

Evaluate lim_(x to 0) (1 - cos ax)/(1 - cos bx)