Question

Find the limit of:
`lim_(x\to 0)(sin2x)/(sin3x)`

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:
`(sin2x)/(sin3x)`

Answer 1

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

`\sf \: lim_(x\to 0) \: (sin2x)/(sin3x)`

• Evaluate the limits of numerator and denominator separately.

`\sf \: lim_(x\to 0) \:( sin(2x)) \\ \sf \:lim_(x\to 0) \:( sin(3x))`

• Evaluate the limit.

`\sf \: 0 \\ \sf \: 0`

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

`\sf \: lim_(x\to 0) \: ((sin(2x))/(sin(3x)))`

• 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...

`\sf \: lim_(x\to 0) \: ((sin(2x) * 2 * 3x)/(sin(3x) * 2 * 3x))`

• Use the commutative property to reorder the terms.

`\sf \: lim_(x\to 0) \: (( 2 * 3x * sin(2x))/(3 * 2x * sin(3x)))`

• Separate the fraction into 3 fractions.

`\sf \: lim_(x\to 0) \: ( (2)/(3) * ( \sin(2x) )/(2x) * (3x)/( \sin(3x) ) )`

• Evaluate the limit.

`\sf \: (2)/(3) * 1 * {1}^( - 1)`

• Simplify the expression.

`\boxed{ \tt (2)/(3)}`