Question #550b2

1 Answer
Jim H
Dec 18, 2016

Please see below.

Explanation:

Use the identities:

#sinA+sinB = 2sin((A+B)/2)cos((A-B)/2)#

#sinA-sinB = 2sin((A-B)/2)cos((A+B)/2)#

to get

#lim_(x rarr 0 ) (sin2x+sin6x)/(sin5x-sin3x) = lim_(xrarr0)(sin6x+sin2x)/(sin5x-sin3x)#

# = lim_(xrarr0)(sin(4x)cos(2x))/(sin(x)cos(4x))#

# = lim_(xrarr0)sin(4x)/sin(x)#

# = 4 lim_(xrarr0)sin(4x)/(4x) (x)/sin(x)#

# = 4*1*1 = 4#

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