How do you evaluate #sin(x-3)/(x^2+4x-21)# as x approaches 3?

Answer 1

#1/10#

#lim_(xrarr3) Sin(x-3)/(x^2+4x-21)#

Using L-Hospital rule,i.e differentiate numerator and denominator separately without using the quotient rule, we get,

#lim_(xrarr3) Cos(x-3)/(2x+4) *d/dx(x-3)#
#lim_(xrarr3) Cos(x-3)/(2x+4)*(1-0)#

Putting x=3,

#Cos(3-3)/{2(3)+4#
=#Cos(0)/10# =#1/10#
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Answer 2

The limit is equal to #1/10#.

To start, let's just try plugging #x=3# into the equation and see what happens:
#color(white)=>sin(x-3)/(x^2+4x-21)#
#=>sin(3-3)/(3^2+4(3)-21)#
#=sin0/(9+12-21)#
#=0/0#
The indeterminate #0/0#. In these types of cases, we can apply l'Hopital's rule:
If #lim_(x->a) f(x)/g(x)=0/0# or #lim_(x->a) f(x)/g(x)=(+- infty)/(+- infty)#, then
#lim_(x->a) f(x)/g(x)=lim_(x->a) (f'(x))/(g'(x))#

In other words, take the derivative of the top and bottom of the fraction, then plug in the value. Let's do that:

#color(white)=lim_(x->3) sin(x-3)/(x^2+4x-21)#
#=lim_(x->3) (d/dx(sin(x-3)))/(d/dx(x^2+4x-21))#
#=lim_(x->3) cos(x-3)/(d/dx(x^2+4x-21))#
#=lim_(x->3) cos(x-3)/(d/dx(x^2)+d/dx(4x)-d/dx(21))#
#=lim_(x->3) cos(x-3)/(2x+4-0)#
#=lim_(x->3) cos(x-3)/(2x+4)#
#=cos(3-3)/(2(3)+4)#
#=cos(0)/(2(3)+4)#
#=cos(0)/(6+4)#
#=cos(0)/10#
#=1/10#

That's the limit. We can observe this from the graph of the function:

graph{sin(x-3)/(x^2+4x-21) [-1, 7, -0.02, 0.2]}

Hope this helped!

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Answer 3

To evaluate sin(x-3)/(x^2+4x-21) as x approaches 3, we can substitute x=3 into the expression. However, this would result in an undefined expression since the denominator would be zero. Therefore, we need to use a different approach. By factoring the denominator, we can rewrite it as (x-3)(x+7). Now, we can cancel out the common factor of (x-3) in the numerator and denominator. This leaves us with sin(0)/(x+7). As x approaches 3, the expression sin(0)/(x+7) simplifies to 0/10, which equals 0. Therefore, the value of sin(x-3)/(x^2+4x-21) as x approaches 3 is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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