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Prove that, underset(x rarr 0)(lim)(5^(x...

Prove that, `underset(x rarr 0)(lim)(5^(x)-4^(x))/(x)=log_(e)((5)/(4))`

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Prove that: underset(x rarr0)lim ((x-1+cosx)/(x))^(1/x) = e^(-1/2)

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CHHAYA PUBLICATION-LIMIT-IIIustrative Examples
  1. Evaluate : underset(xrarr1)"lim"(e^(logx)-1)/(e^(x-1)-1)

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  2. Evaluate : underset(xrarr2)"lim"(e^(x)-e^(2))/(x-2)

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  3. Prove that, underset(x rarr 0)(lim)(5^(x)-4^(x))/(x)=log(e)((5)/(4))

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  4. Evaluate: underset(x rarr 0)(lim)log(1+alphax)/(e^(2x)-1)

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  5. Evaluate : underset(xrarr0)"lim"(e^(ax)-1)/(sinbx)

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  6. Evaluate : underset(xrarr0)"lim"(sinx)/(log(e)(1+x)^(1//4))

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  7. Evaluate : underset(xrarr0)"lim"((e^(x)-1)log(1+x))/(sin^(2)x)

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  8. Evaluate : underset(xrarr0)"lim"(3^(x)-1)/((4+x)^(1//2)-2)

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  9. Evaluate :underset(nrarroo)"lim"(n^(4)-n^(2)+6n+1)/(2n^(4)-5n+4)

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  10. Evaluate :underset(xrarroo)"lim"(sqrt(x^(2)+ax)-x).

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  11. Prove that underset(nrarroo)"lim"(1^(3)+2^(3)+3^(3)+...+n^(3))/(n^(4))...

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  12. Show that underset(xrarr0)"lim"(xtan2x-2xtanx)/((1-cos2x)^(2))=(1)/(2)

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  13. Examine the existence of the following limits : underset(xrarr2)"li...

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  14. Examine the existence of the following limits : underset(xrarr3)"lim...

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  15. Prove that underset(xrarr0)"lim"(3x+|x|)/(7x-5|x|)does not exist .

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  16. Does underset(xrarra)"lim"f(x)=underset(xrarra)"lim"phi(x) "always imp...

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  17. Examine the existence of the following of limits : underset(xrarr...

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  18. Examine the existence of the following of limits : underset(xrarr0)"l...

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  19. If underset(xrarr0)"lim"(1+x)^(1/x)=e "prove that "underset(xrarr0)"li...

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  20. A function f (x) is defined as follwos : f(x){:(=2x+1," when "xle...

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