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x → ∞ lim 5 x − 1 + 10 0 2 x + 2 + 4000 2 5 x − 1 0 ( x + 1 ) + 80000 = ...

$x \to \infty lim \frac{2 5 ^{x} - 1 0 ^{(x + 1)} + 80000}{5 ^{x - 1} + 10 0 ^{\frac{x + 2}{2}} + 4000} = ...$

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$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of fraction numerator square root of 25 to the power of straight x end root minus 10 to the power of open parentheses straight x plus 1 close parentheses end exponent plus 80000 over denominator 5 to the power of straight x minus 1 end exponent plus 100 to the power of begin display style fraction numerator straight x plus 2 over denominator 2 end fraction end style end exponent plus 4000 end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator 5 to the power of straight x minus 10 to the power of straight x plus 1 end exponent plus 80000 over denominator 5 to the power of straight x minus 1 end exponent plus 100 to the power of begin display style straight x plus 2 end style end exponent plus 4000 end fraction. fraction numerator begin display style 1 over 10 to the power of straight x end style over denominator 1 over 10 to the power of straight x end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator open parentheses begin display style 5 over 10 end style close parentheses to the power of straight x minus 10 plus begin display style 80000 over 10 to the power of straight x end style over denominator begin display style 1 fifth end style open parentheses 5 over 10 close parentheses to the power of straight x plus 100 plus 4000 over 10 to the power of straight x end fraction end cell row blank equals cell fraction numerator 0 minus 10 plus 0 over denominator 0 plus 100 plus 0 end fraction end cell row blank equals cell negative 1 over 10 end cell row blank equals cell negative 0 comma 1 end cell end table end style$

Maka $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator square root of 25 to the power of straight x end root minus 10 to the power of open parentheses straight x plus 1 close parentheses end exponent plus 80000 over denominator 5 to the power of straight x minus 1 end exponent plus 100 to the power of begin display style fraction numerator straight x plus 2 over denominator 2 end fraction end style end exponent plus 4000 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 0 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table end style$