Limits Involving Trigonometric Functions
Key Questions

#lim_(x>0) (cos(x)1)/x = 0# . We determine this by utilising L'hospital's Rule.To paraphrase, L'Hospital's rule states that when given a limit of the form
#lim_(x→a)f(x)/g(x)# , where#f(a)# and#g(a)# are values that cause the limit to be indeterminate (most often, if both are 0, or some form of ∞), then as long as both functions are continuous and differentiable at and in the vicinity of#a,# one may state that#lim_(x→a)f(x)/g(x)=lim_(x→a)(f'(x))/(g'(x))# Or in words, the limit of the quotient of two functions is equal to the limit of the quotient of their derivatives.
In the example provided, we have
#f(x)=cos(x)1# and#g(x)=x# . These functions are continuous and differentiable near#x=0, cos(0) 1 =0 and (0)=0# . Thus, our initial#f(a)/g(a)=0/0=?.# Therefore, we should make use of L'Hospital's Rule.
#d/dx (cos(x) 1)=sin(x), d/dx x=1# . Thus...#lim_(x>0) (cos(x)1)/x = lim_(x>0)(sin(x))/1 = sin(0)/1 = 0/1 = 0# 
#lim_(x>0) sin(x)/x = 1# . We determine this by the use of L'Hospital's Rule.To paraphrase, L'Hospital's rule states that when given a limit of the form
#lim_(x>a) f(x)/g(x)# , where#f(a)# and#g(a)# are values that cause the limit to be indeterminate (most often, if both are 0, or some form of#oo# ), then as long as both functions are continuous and differentiable at and in the vicinity of#a# , one may state that#lim_(x>a) f(x)/g(x) = lim_(x>a) (f'(x))/(g'(x))# Or in words, the limit of the quotient of two functions is equal to the limit of the quotient of their derivatives.
In the example provided, we have
#f(x) = sin(x)# and#g(x) = x# . These functions are continuous and differentiable near#x=0# ,#sin(0) = 0# and#(0) = 0# . Thus, our initial#f(a)/g(a) = 0/0 = ?# . Therefore, we should make use of L'Hospital's Rule.#d/dx sin(x) = cos(x), d/dx x = 1# . Thus...#lim_(x>0) sin(x)/x = lim_(x>0) cos(x)/1 = cos(0)/1 = 1/1 = 1#
Questions
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