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Tentukan nilai limit berikut.

b)  limit as x rightwards arrow infinity of space open parentheses square root of open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses end root minus open parentheses x square root of 2 plus 1 close parentheses close parentheses

S. Dwi

Master Teacher

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Jawaban

 limit as x rightwards arrow infinity of space open parentheses square root of open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses end root minus open parentheses x square root of 2 plus 1 close parentheses close parentheses 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 cell fraction numerator 3 square root of 2 over denominator 4 end fraction minus 1 end cell end table.

Pembahasan

Perhatikan perhitungan berikut!

dengan substitusi langsung:

limit as x rightwards arrow infinity of space open parentheses square root of open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses end root minus open parentheses x square root of 2 plus 1 close parentheses close parentheses equals infinity minus infinity space open parentheses tak space tentu close parentheses 

karena dengan substitusi langsung hasilnya tak tentu, maka dilakukan perhitungan sebagai berikut:

Ingat rumus berikut ini!

Jikalimit as x rightwards arrow infinity of square root of a x squared plus b x plus c end root minus square root of p x squared plus q x plus r end root, maka diperoleh:

  1. L equals negative infinity space, jika a less than p
  2. L equals fraction numerator b minus q over denominator 2 square root of a end fraction, jika a equals p
  3. L equals infinity, jika a greater than p

L adalah nilai limit

Dari rumus tersebut, perhatikan perhitungan berikut!

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of space open parentheses square root of open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses end root minus open parentheses x square root of 2 plus 1 close parentheses close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of space open parentheses square root of open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses end root minus square root of open parentheses x square root of 2 plus 1 close parentheses squared end root close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses square root of 2 x squared plus 3 x minus 2 end root minus square root of 2 x squared plus 2 square root of 2 plus 1 end root close parentheses end cell row blank equals cell fraction numerator 3 minus 2 square root of 2 over denominator 2 square root of 2 end fraction end cell row blank equals cell fraction numerator 3 minus 2 square root of 2 over denominator 2 square root of 2 end fraction times fraction numerator 2 square root of 2 over denominator 2 square root of 2 end fraction end cell row blank equals cell fraction numerator 6 square root of 2 minus 8 over denominator 8 end fraction end cell row blank equals cell fraction numerator 3 square root of 2 over denominator 4 end fraction minus 1 end cell end table

Diperoleh a equals p equals 2, maka menghitung nilai limitnya menggunakan cara yang nomer 2

Dengan demikian, limit as x rightwards arrow infinity of space open parentheses square root of open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses end root minus open parentheses x square root of 2 plus 1 close parentheses close parentheses 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 cell fraction numerator 3 square root of 2 over denominator 4 end fraction minus 1 end cell end table.

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