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Show that Lt(xto0)(a^(x)-1)/(b^(x)-1)=lo...

Show that `Lt_(xto0)(a^(x)-1)/(b^(x)-1)=log_(b)a` where a and b are positive numbers other than unity.

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The correct Answer is:
`log_(b)a`
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AAKASH SERIES-LIMITS-SOLVED EXAMPLES
  1. Show that Lt(xto0)(e^(ax)-1)/x=a(ainR)

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  2. Prove that Lt(x to 0) (a^(kx)-1)/(x)=k. log a (a gt theta, a ne 1)

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  3. Show that Lt(xto0)(a^(x)-1)/(b^(x)-1)=log(b)a where a and b are positi...

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  4. Evulate Lt(x to 0)(27^(x)-9^(x)-3^(x)+1)/(x^(2))

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  5. Evaluate Lt(xto0)((e^(x^(2))-cosx)/x^(2)).

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  6. underset(n to oo)lim (n(1^(3)+2^(3)+...+n^(3))^(2))/((1^(2)+2^(2)+...+...

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  7. underset(n to oo)lim {(1)/(1.4)+(1)/(4.7)+(1)/(7.10)+....+(1)/((3n-2)(...

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  8. Show that Lt(ntooo)(3.2^(n+1)-4.5^(n+1))/(5.2^(n)+7.5^(n))=-20/7

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  9. Lt(ntooo)((n+2)!+(n+1)!)/((n+2)!-(n+1)!)

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  10. Evaulate Lt(x to 0)(1+ax)^(b//x) where a,b are constants.

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  11. Evaluate Lt(xtooo)(1+1/(ax+b))^(cx+d) where a, b, c, d are positive co...

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  12. Find Lt(x to 0)((sinx)/(x))^((sinx)/(x-sinx))

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  13. Find Lt(xtooo)((x+a)/(x+b))^(x), a, b are constants.

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  14. Evaluate Lt(xtooo)((x^(2)+ax+b)/(x^(2)+cx+d))^(x) where a, b, c, d are...

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  15. Find Lt(x to oo)((3x^(2)+1)/(4x^(2)-1))^((x^(3))/(1+x)) Caution : Th...

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  16. Evaluate Lt(xto0)((a^(x)+b^(x)+c^(x))/3)^(1/x) where a,b,cinR^(+).

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  17. If Lt(x to 0)(1+ax+bx^(2))^(2//x)=e^(3), find an and b.

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  18. Find Lt(xto0)((sinx-x+x^(3)/6)/x^(5))

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  19. Find a, b and c if Lt(xto0)((ae^(x)-bcosx+ce^(-x))/x^(2))=2

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  20. Evaluate Lt(x to 0)[((1+x)^((1)/(x))-e+(ex)/(2))/(sin^(2)x)]

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