What is the equation of the tangent line of #f(x)=sqrt(x+1)/sqrt(x+2) # at #x=4#?

Answer 1

Equation of tangent is #x-12sqrt30y+56=0#

Thr slope of a tangent to a curve #f(x)# at a point #x=x_0# is given by value of its derivative at #x=x_0# i.e. #f'(x_0)#.
As #f(x)=sqrt(x+1)/sqrt(x+2)#
Using quotient rule, #f'(x)=(sqrt(x+2)xx1/(2sqrt(x+1))-sqrt(x+1)xx1/(2sqrt(x+2)))/(x+2)#
and #f'(4)=(sqrt6/(2sqrt5)-sqrt5/(2sqrt6))/6#
= #(1/sqrt30)/12=1/(12sqrt30)#
Now at #x=4#, #f(x)=sqrt5/sqrt6#.

Hence, using point slope form, equation of desired tangent is

#y-sqrt5/sqrt6=1/(12sqrt30)(x-4)#
or #12sqrt30y-60=x-4#
or #x-12sqrt30y+56=0#

graph{(y-sqrt(x+1)/sqrt(x+2))(x-12sqrt30y+56)=0 [-0.29, 4.71, -0.53, 1.97]}

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Answer 2

To find the equation of the tangent line of f(x) = sqrt(x+1)/sqrt(x+2) at x=4, we need to find the derivative of f(x) and evaluate it at x=4. The derivative of f(x) can be found using the quotient rule:

f'(x) = [sqrt(x+2) * (1/2) * (x+1)^(-1/2) - sqrt(x+1) * (1/2) * (x+2)^(-1/2)] / (x+2)

Evaluating f'(x) at x=4, we get:

f'(4) = [sqrt(4+2) * (1/2) * (4+1)^(-1/2) - sqrt(4+1) * (1/2) * (4+2)^(-1/2)] / (4+2)

Simplifying this expression, we find:

f'(4) = [sqrt(6) * (1/2) * (5)^(-1/2) - sqrt(5) * (1/2) * (6)^(-1/2)] / 6

Simplifying further, we get:

f'(4) = (sqrt(6) / (2 * sqrt(5))) - (sqrt(5) / (2 * sqrt(6))) / 6

Combining the terms, we have:

f'(4) = (sqrt(6) * sqrt(6) - sqrt(5) * sqrt(5)) / (2 * sqrt(5) * sqrt(6)) / 6

Simplifying the numerator, we get:

f'(4) = (6 - 5) / (2 * sqrt(5) * sqrt(6)) / 6

Simplifying further, we have:

f'(4) = 1 / (2 * sqrt(5) * sqrt(6)) / 6

Therefore, the equation of the tangent line of f(x) at x=4 is:

y = f(4) + f'(4) * (x - 4)

Substituting the values of f(4) and f'(4) into the equation, we get:

y = sqrt(4+1)/sqrt(4+2) + 1 / (2 * sqrt(5) * sqrt(6)) / 6 * (x - 4)

Simplifying this equation, we have:

y = sqrt(5)/sqrt(6) + 1 / (12 * sqrt(5) * sqrt(6)) * (x - 4)

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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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