Pertanyaan

Tentukanlah: x → 0 lim ( x 2 − 3 x + 2 2 x 2 − 2 x + 3 ) x s i n 2 x

Tentukanlah:

$x \to 0 lim (\frac{2 x ^{2} - 2 x + 3}{x ^{2} - 3 x + 2})^{\frac{s i n 2 x}{x}}$

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Pembahasan

Ingat kembali rumus dan sifat berikut. Berdasarkan rumus di atas, diperoleh perhitungan sebagai berikut. Jadi, hasil dari adalah .

Ingat kembali rumus dan sifat berikut.

$limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator b x end fraction equals a over b$

Berdasarkan rumus di atas, diperoleh perhitungan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of open parentheses fraction numerator 2 x squared minus 2 x plus 3 over denominator x squared minus 3 x plus 2 end fraction close parentheses to the power of fraction numerator sin space 2 x over denominator x end fraction end exponent end cell equals cell limit as x rightwards arrow 0 of open parentheses fraction numerator 2 x squared minus 2 x plus 3 over denominator x squared minus 3 x plus 2 end fraction close parentheses to the power of limit as x rightwards arrow 0 of fraction numerator sin space 2 x over denominator x end fraction end exponent end cell row blank equals cell limit as x rightwards arrow 0 of open parentheses fraction numerator 2 x squared minus 2 x plus 3 over denominator x squared minus 3 x plus 2 end fraction close parentheses to the power of 2 over 1 end exponent end cell row blank equals cell limit as x rightwards arrow 0 of open parentheses fraction numerator 2 x squared minus 2 x plus 3 over denominator x squared minus 3 x plus 2 end fraction close parentheses squared end cell row blank equals cell open parentheses fraction numerator 2 left parenthesis 0 right parenthesis squared minus 2 left parenthesis 0 right parenthesis plus 3 over denominator left parenthesis 0 right parenthesis squared minus 3 left parenthesis 0 right parenthesis plus 2 end fraction close parentheses squared end cell row blank equals cell left parenthesis 3 over 2 right parenthesis squared space end cell row blank equals cell 9 over 4 end cell end table$

Jadi, hasil dari $limit as x rightwards arrow 0 of open parentheses fraction numerator 2 x squared minus 2 x plus 3 over denominator x squared minus 3 x plus 2 end fraction close parentheses to the power of fraction numerator sin space 2 x over denominator x end fraction end exponent$ adalah 9 over 4 .