Limits Involving Trigonometric Functions
Key Questions
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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) , wheref(a) andg(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 ofa, one may state thatlim_(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 andg(x)=x . These functions are continuous and differentiable nearx=0, cos(0) -1 =0 and (0)=0 . Thus, our initialf(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) , wheref(a) andg(a) are values that cause the limit to be indeterminate (most often, if both are 0, or some form ofoo ), then as long as both functions are continuous and differentiable at and in the vicinity ofa , one may state thatlim_(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) andg(x) = x . These functions are continuous and differentiable nearx=0 ,sin(0) = 0 and(0) = 0 . Thus, our initialf(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
Differentiating Trigonometric Functions
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Limits Involving Trigonometric Functions
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Intuitive Approach to the derivative of y=sin(x)
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Derivative Rules for y=cos(x) and y=tan(x)
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Differentiating sin(x) from First Principles
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Special Limits Involving sin(x), x, and tan(x)
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Graphical Relationship Between sin(x), x, and tan(x), using Radian Measure
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Derivatives of y=sec(x), y=cot(x), y= csc(x)
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Differentiating Inverse Trigonometric Functions