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Find the limit of: [tex]lim_{x\to 0}\frac{sin2x}{sin3x}[/tex]

Please use the hint in the problem, but also explain what the hint is. I don't know how they derived it from the original equation of: [tex]\frac{sin2x}{sin3x}[/tex]

Find the limit of texlimxto 0fracsin2xsin3xtex Please use the hint in the problem but also explain what the hint is I dont know how they derived it from the ori class=

Respuesta :

itzadrien

First of all I am solving the question and then I will explain the hint...

[tex] \sf \: lim_{x\to 0} \: \frac{sin2x}{sin3x}[/tex]

  • Evaluate the limits of numerator and denominator separately.

[tex] \sf \: lim_{x\to 0} \:( sin(2x)) \\ \sf \:lim_{x\to 0} \:( sin(3x))[/tex]

  • Evaluate the limit.

[tex] \sf \: 0 \\ \sf \: 0[/tex]

  • Since the expression 0/0 is an indeterminate form, try transforming the expression.

[tex] \sf \: lim_{x\to 0} \: (\frac{sin(2x)}{sin(3x)})[/tex]

  • Multiply the fraction by 2×3x/2×3x
  • Now, Here we will make the use of hint.. When we evaluated the limit we got 0/0 so now we will multiply the fraction by 2×3x because we need to simplify or we can say eationalize the denominator...

[tex] \sf \: lim_{x\to 0} \: (\frac{sin(2x) \times 2 \times 3x}{sin(3x) \times 2 \times 3x})[/tex]

  • Use the commutative property to reorder the terms.

[tex] \sf \: lim_{x\to 0} \: (\frac{ 2 \times 3x \times sin(2x)}{3 \times 2x \times sin(3x)})[/tex]

  • Separate the fraction into 3 fractions.

[tex] \sf \: lim_{x\to 0} \: ( \frac{2}{3} \times \frac{ \sin(2x) }{2x} \times \frac{3x}{ \sin(3x) } )[/tex]

  • Evaluate the limit.

[tex] \sf \: \frac{2}{3} \times 1 \times {1}^{ - 1} [/tex]

  • Simplify the expression.

[tex] \boxed{ \tt \frac{2}{3}}[/tex]

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