[ASK] Limit of Trigonometry Function

Click For Summary
SUMMARY

The limit of the trigonometric function $$\lim_{x\to0}\frac{sin2x+sin6x+sin10x-sin18x}{3sinx-sin3x}$$ evaluates to 192. The discussion highlights that both substitution and L'Hôpital's Rule initially yield an indeterminate form of $$\frac{0}{0}$$. By applying L'Hôpital's Rule multiple times, specifically until the third derivative, the limit simplifies to $$\frac{4,608}{24}$$, confirming the result as 192.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with L'Hôpital's Rule
  • Knowledge of trigonometric functions and their derivatives
  • Ability to compute derivatives up to the third order
NEXT STEPS
  • Study advanced applications of L'Hôpital's Rule in calculus
  • Learn about Taylor series expansions for trigonometric functions
  • Explore the concept of indeterminate forms in limits
  • Practice solving limits involving trigonometric functions
USEFUL FOR

Students and educators in calculus, mathematicians focusing on limits, and anyone seeking to deepen their understanding of trigonometric limits and derivatives.

Monoxdifly
MHB
Messages
288
Reaction score
0
$$\lim_{x\to0}\frac{sin2x+sin6x+sin10x-sin18x}{3sinx-sin3x=}$$
A. 0
B. 45
C. 54
D. 192
E. 212

Either substituting or using L'Hopital gives $$\frac00$$. Is there any way to simplify it and make the result a real number?
 
Physics news on Phys.org
We can repeat L'Hôpital until it's not $\frac 0 0$ any more.
 
Ah, I see. It's in the third derivative and the answer is $$\frac{4,608}{24}$$= 192, right?
 
Monoxdifly said:
Ah, I see. It's in the third derivative and the answer is $$\frac{4,608}{24}$$= 192, right?
Yep. (Nod)
 

Similar threads

MHB 1.4.203 AP calculus practice question on Limits
  • · Replies 1 ·
Replies
1
Views
2K
MHB Limit Function: Finding Solution
  • · Replies 3 ·
Replies
3
Views
1K
MHB Finding a limit of multivariable function
  • · Replies 2 ·
Replies
2
Views
2K
MHB Evaluating limit limit of multivariable function
  • · Replies 3 ·
Replies
3
Views
2K
MHB Limit $\frac{f(x)}{g(x)}$: Solve w/ L'H Rule
  • · Replies 4 ·
Replies
4
Views
2K
MHB Calculating Limits without Cheating or L'Hopital's Rule
  • · Replies 5 ·
Replies
5
Views
3K
MHB 2.1.207 AP calculus practice exam problem Lim with tan(4X)/6x
  • · Replies 2 ·
Replies
2
Views
2K
MHB Simplifying Limit Problem - L'Hôpital and Algebra
  • · Replies 2 ·
Replies
2
Views
3K
Limit of a Function at a Specific Value
  • · Replies 10 ·
Replies
10
Views
2K
Solve Limit: $$\lim_{x\to0} \frac{sin(\pi(Cos^2(x)))}{\pi(Cos^2(x))}$$
  • · Replies 5 ·
Replies
5
Views
2K