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Hitunglah nilai limit di bawah ini!

begin mathsize 14px style limit as x rightwards arrow infinity of space open parentheses open parentheses x plus 2 close parentheses minus square root of x squared minus 6 x plus 10 end root close parentheses end style 

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Jawaban

nilai dari begin mathsize 14px style limit as x rightwards arrow infinity of space open parentheses open parentheses x plus 2 close parentheses minus square root of x squared minus 6 x plus 10 end root close parentheses end style adalah 5.

Pembahasan

Langkah pertama dalam menyelesaikan limit tak hingga bentuk akar adalah dengan mengalikan akar sekawan. Selanjutnya, membagi semua suku pembilang dan penyebut dengan variabel yang memiliki pangkat tertinggi. Maka diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of space open parentheses open parentheses x plus 2 close parentheses minus square root of x squared minus 6 x plus 10 end root close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of space open parentheses open parentheses x plus 2 close parentheses minus square root of x squared minus 6 x plus 10 end root close parentheses end cell row blank blank cell cross times fraction numerator open parentheses open parentheses x plus 2 close parentheses plus square root of x squared minus 6 x plus 10 end root close parentheses over denominator open parentheses open parentheses x plus 2 close parentheses plus square root of x squared minus 6 x plus 10 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator open parentheses x plus 2 close parentheses squared minus open parentheses square root of x squared minus 6 x plus 10 end root close parentheses squared over denominator open parentheses x plus 2 close parentheses plus square root of x squared minus 6 x plus 10 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator open parentheses x squared plus 4 x plus 4 close parentheses minus open parentheses x squared minus 6 x plus 10 close parentheses over denominator open parentheses x plus 2 close parentheses plus square root of x squared minus 6 x plus 10 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator 10 x minus 6 over denominator open parentheses x plus 2 close parentheses plus square root of x squared minus 6 x plus 10 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator begin display style fraction numerator 10 x over denominator x end fraction end style minus begin display style 6 over x end style over denominator open parentheses begin display style x over x end style plus begin display style 2 over x end style close parentheses plus square root of begin display style x squared over x squared end style minus begin display style fraction numerator 6 x over denominator x squared end fraction end style plus begin display style 10 over x squared end style end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator begin display style 10 minus 6 over x end style over denominator open parentheses begin display style 1 plus 2 over x end style close parentheses plus square root of begin display style 1 minus 6 over x plus 10 over x squared end style end root end fraction end cell row blank equals cell fraction numerator 10 minus 0 over denominator open parentheses 1 plus 0 close parentheses plus square root of 1 minus 0 minus 0 end root end fraction end cell row blank equals cell 10 over 2 end cell row blank equals 5 end table end style 

Jadi, nilai dari begin mathsize 14px style limit as x rightwards arrow infinity of space open parentheses open parentheses x plus 2 close parentheses minus square root of x squared minus 6 x plus 10 end root close parentheses end style adalah 5.

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