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lim(m to oo)("cos"(x)/(m))^(m) is equal ...

`lim_(m to oo)("cos"(x)/(m))^(m)` is equal to

A

0

B

e

C

`(1)/(e )`

D

1

Text Solution

Verified by Experts

The correct Answer is:
D
`lim_(m to oo) ("cos"(x)/(m))^(m)=lim_(m to oo) [1+("cos" (x)/(m)-1)]^(m)`
`=lim_(m to oo)[1-(1-"cos"(x)/(m))]^(m)`
` =lim_(m to oo) [1-2"sin"^(2)(x)/(2m)]^(m)=e^(lim_(m to oo) m[(-2"sin"^(2)(x)/(2m)])`
`=e^(lim_(m to oo) -2(("sin"(x)/(2m))/(x//2m))^(2)((x^(2))/(4m^(2)))m)`
`=e^(-2 lim_( m to oo) (x^(2))/(4m))=e^(0)=1`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-CONTINUITY-EXERCISE 2 (MISCELLANEOUS PROBLEMS)
  1. lim(x to0) (-1)^([x]) where [.] denotes the greatest function is equal...

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  2. If lim(x to 1) (ax^(2)+bx+c)/((x-1)^(2))=2, then (a, b, c) is

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  3. lim(m to oo)("cos"(x)/(m))^(m) is equal to

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  4. The value of lim( x to 3) (x^(5)-3^(5))/(x^(8)-3^(8)) is equal to

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  5. lim(x to 0) ((1-cos 2x)(3+cosx))/(x tan 4x) is equal to

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  6. At x=3, f(x) ={(x^(5)-243",", "if " x ne 3),(x^(3)-27",", "if " x=3):}...

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  7. The value of f(0), if f(x)=(x tan 2x)/(sin3x sin5x) is continuous at ...

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  8. f(x)={((5^(cosx)-1)/((pi)/(2)-x)",", x ne (pi)/(2)), (log 5"," , x =(p...

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  9. If f(x)={((sqrt(1+px)-sqrt(1-px))/(x) ",", -1 le x lt 0),((2x+1)/(x-2...

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  10. The function f(x)=(x^3+x^2-16 x+20)/(x-2) is not defined for x=2. In o...

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  11. In order that the function f(x)=(x+1)^(cotx) is continuous at x = 0, t...

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  12. if f(x) is continuous and f(9/2)=2/9 then the value if lim(x->0)f((1-c...

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  13. If f(x)={(ax+1",", x le (pi)/(2)),(sinx+b",", x gt (pi)/(2)):} is cont...

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  14. If f(x)={((1+2x)^(1//x)",","for " x ne 0),(e^(2)",", "for " x =0):}, t...

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  15. If f(y)={(((e^(2y)-1)*siny)/(y^(2)),", for "y ne 0),(4,", for "y=0):}...

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  16. If f(x)={((tanx)/(sinx)",", x ne 0),(1",", x =0):} then f(x) is

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  17. For the function f(x)={((sin^(2)ax)/(x^(2))",","where " x ne 0),(1",",...

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  18. If f(x)={(x+a sqrt(2) sinx"," ,0 lt x lt (pi)/(4)),(2x cotx+b",",(pi)...

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  19. Which of the following functions is continuous at x = 0 ?

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  20. For what value of k, function f(x)={((k cosx)/(pi-2x)",","if "x ne (pi...

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