Find limit as x->-3 of (sin(4x+12))/(5x+15)

Answer 1

limit (sin(4x+12))/(5x+15) as x -> -3
If we try substituting,
= (sin(4(-3)+12))/(5(-3)+15)
= sin(-12+12) / -15+15
= sin(0) / 0
= 0 /0
Note that 0/0 is indeterminate, and if you arrive at this answer, you use the L'hopital's rule.
To use the L'hopital's rule, get the derivative of numerator and denominator separately, then substitute x. Here,
derivative of sin(4x+12) = 4*cos(4x+12)
derivative of 5x+15 = 5
Rewriting,
limit 4*cos(4x+12)/(5) as x -> -3
= 4*cos(-12+12) / 5
= 4*cos(0) / 5
= 4*1 / 5
= 4/5

If however, after using L'hopital's rule, you arrive at 0/0 answer again, use the rule again, until the answer is different.
Hope this helps :)

Answer 2

That helped a whole lot! Thank you so much!