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Show that ("lim")(x->0)x/(|x|) does not ...

Show that `("lim")_(x->0)x/(|x|)` does not exist.

Text Solution

Verified by Experts

Let `f(x)=(x)/(|x|).` Then,
`underset(xto0^(+))limf(x)=underset(hto0)limf(0+h)=underset(hto0)limf(h)=underset(hto0)lim(h)/(|h|)=underset(hto0)limh/h=1.`
`underset(xto0^(-))limf(x)underset (xto0)limf(0-h)=underset(hto0)limf(-h)=underset(hto0)lim(-h)/(|-h|)=underset(hto0)lim(-h)/(h)=-1.`
`thereforeunderset(xto0^(+))limf(x)neunderset(xto0^(-))limf(x).`
Hence, `underset(xto0)limf(x)` does not exist.
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RS AGGARWAL-LIMIT-SOLVED EXAMPLES
  1. Evaluate lim(xrarr0)((x^(3)cotx)/(1-cosx)).

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  2. lim(x->0)(sin3x+7x)/(4x+sin2x)

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  3. Evaluate lim(x-> 0) (tan3x-2x)/(3x- sin^2 x)

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  4. Evaluate lim(xrarr0)(xtan4x)/(1-cos4x).

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  5. Evaluate lim(xrarr0)((1-cosxsqrt(cos2x)))/(x^(2)).

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  6. Evaluate lim(xrarr(pi)/(4))((sinx-cosx))/((x-(pi)/(4))).

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  7. Evaluate lim(xrarr(pi)/(2))(cosx)/(((pi)/(2)-x)).

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  8. Evaluate lim(xrarr(pi)/(6))((sqrt3sinx-cosx))/((x-(pi)/(6))).

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  9. Let f(x){:{(2x+3",",xle0),(3(x+1)",",xgt0.):} Find (i) lim(xrarr0)f(...

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  10. Let f(x)={:{(x^(2)-1",",xle1),(-x^(2)-1",",xgt1.):} "Find"lim(xrarr1)f...

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  11. Let f(x)={:{((|x|)/(x)",",xne0),(0",",x=0.):} Find lim(xrarr0)f(x).

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  12. Suppose f(x)={(a+bx, x<1), (4, x=1), (b-ax, x>1):} and if lim(xrarr1) ...

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  13. If f(x)={{:(|x|+1,xlt0),(0,x=0) ,(|x|-1,xgt0):} for what value (s) ...

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  14. Let f(x)={{:(mx^(2)+n",",xlt0),(nx+m",",0lexle1),(nx^(3)+m",",xgt1):} ...

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  15. Let a(1),a(2),...,a(n) be fixed real numbers and let f(x)=(x-a(1))(...

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  16. Let f(x)={x}= greater integer less than or eqal to x. For any integer ...

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  17. If is an odd function and underset(xto0)f(x) exists then prove tha...

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  18. If f is an even function, prove that lim(xto0^(-)) (x)=lim(xto0^(+)) ...

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  19. Show that lim(xto0^(-)) ((e^(1//x)-1)/(e^(1//x)+1)) does not exist.

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  20. Show that ("lim")(x->0)x/(|x|) does not exist.

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