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Evaluate: lim(x->0)(10^x-2^x-5^x+1)/(xta...

Evaluate: `lim_(x->0)(10^x-2^x-5^x+1)/(xtanx)`

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Verified by Experts

We have
`underset(xto0)lim(10^(x)-2^(x)-5^(x)+1)/(xtanx)" "`(0/0 from)
`=underset(xto0)lim(5^(x).2^(x)-2^(x)-5^(x)+1)/(xtanx)`
`=underset(xto0)lim((5^(x)-1)(2^(x)-1))/(xtanx)`
`=underset(xto0)lim(5^(x)-1)/(x)(2^(x)-1)/(x)(x)/(tanx)`
`=underset(xto0)lim(5^(x)-1)/(x)underset(xto0)lim(2^(x)-1)/(x)underset(xto0)lim(x)/(tanx)`
=(log5) (log2) (1)
(log5)(log2)
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