Question #a1349 Calculus Differentiating Trigonometric Functions Limits Involving Trigonometric Functions 1 Answer Cesareo R. Apr 6, 2017 -8 Explanation: Applying l'Hopital's rule lim_(x->0)(-5sin(5x)+3sin(3x))/(2x)=1/2lim_(x->0)(-5sin(5x)/x+3sin(3x)/x)= =1/2lim_(x->0)(-5^2sin(5x)/(5x)+3^2sin(3x)/(3x)) but lim_(y->0)sin(y)/y = 1 so lim_(x->0)(cos(5x)-cos(3x))/x^2 = =1/2lim_(x->0)(-5^2sin(5x)/(5x)+3^2sin(3x)/(3x))=1/2(3^2-5^2)=-8 Answer link Related questions How do you find the limit of inverse trig functions? How do you find limits involving trigonometric functions and infinity? What is the limit lim_(x->0)sin(x)/x? What is the limit lim_(x->0)(cos(x)-1)/x? What is the limit of sin(2x)/x^2 as x approaches 0? Question #99ee1 What is the derivative of 2^sin(pi*x)? What is the derivative of sin^3x? Question #eefeb Question #af14f See all questions in Limits Involving Trigonometric Functions Impact of this question 1939 views around the world You can reuse this answer Creative Commons License