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If lim(xtoa) {(f(x))/(g(x))} exists, the...

If `lim_(xtoa) {(f(x))/(g(x))}` exists, then which one of the following correct ?

A

both `underset(xtoa)limf(x)` and `underset(xtoa)limg(x)` must exist

B

`underset(xtoa)limf(x)`need not exist `underset(xtoa)limg(x)` exist

C

neither `underset(xtoa)limf(x)` nor `underset(xtoa)limg(x)` may exist

D

`underset(xtoa)limf(x)` exist but `underset(xtoa)limg(x)` need not exist

Text Solution

Verified by Experts

The correct Answer is:
C
If `f(x)=sin((1)/(x))" and "g(x)=(1)/(x)," then both "underset(xto0)limf(x)" and "underset(xto0)limg(x)" do nto exist, but "underset(xto0)lim(f(x))/(g(x))=0" exists"`
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CENGAGE ENGLISH-LIMITS-Exercises (Single Correct Answer Type)
  1. lim(xto1) [cosec(pix)/(2)]^(1//(1-x)) (where [.] represents the greate...

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  2. The value of the limit lim(xto0) (a^(sqrt(x))-a^(1//sqrt(x)))/(a^(sqrt...

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  3. If lim(xtoa) {(f(x))/(g(x))} exists, then which one of the following c...

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  4. lim(xto-1) (1)/(sqrt(|x|-{-x})) (where {x} denotes the fractional part...

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  5. If x(1)=3 and x(n+1)=sqrt(2+x(n))" ",nge1, then lim(ntooo) x(n)is

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  6. lim(xto0^(-)) (sum(r=1)^(2n+1)[x^(r)]+(n+1))/(1+[x]+|x|+2x), where nin...

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  7. lim(xtooo) (sin^(4)x-sin^(2)x+1)/(cos^(4)x-cos^(2)x+1)is equal to

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  8. If f(x)=(2)/(x-3),g(x)=(x-3)/(x+4)," and "h(x)=-(2(2x+1))/(x^(2)+x-12)...

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  9. The value of lim(xto pi) (1+cos^(3)x)/(sin^(2)x)" is "

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  10. The value of lim(xto2) (sqrt(1+sqrt(2+x))-sqrt(3))/(x-2)" is "

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  11. The value of lim(xto2) (2^(x)+2^(3-x)-6)/(sqrt(2^(-x))-2^(1-x))" is "

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  12. The value of lim(xto2) (((x^(3)-4x)/(x^(3)-8))^(-1)-((x+sqrt(2x))/(x-2...

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  13. If lim(xto-2^(-)) (ae^(1//|x+2|)-1)/(2-e^(1//|x+2|))=lim(xto-2^(+)) si...

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  14. lim(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^...

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  15. The value of lim(xto(1)/(sqrt(2))) (x-cos(sin^(-1)x))/(1-tan(sin^(-1)x...

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  16. Among (i) lim(xtooo) sec^(-1)((x)/(sinx))" and "(ii) lim(xtooo) sec^(-...

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  17. lim(xtooo) ((x^(3))/(3x^(2)-4)-(x^(2))/(3x+2))" is equal to "

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  18. lim(ntooo) (n(2n+1)^(2))/((n+2)(n^(2)+3n-1))" is equal to "

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  19. lim(xtooo) ((2x+1)^(40)(4x+1)^(5))/((2x+3)^(45)) is equal to

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  20. lim(xtooo) [sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)] is equal to

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