Find an equation of the tangent line to the graph of f at the given point.

f(x) = x + (4/x), (4,5)
I tried using the limit process a lot, but it didn't quite work. If possible, please use the limit process. Thanks a lot to whoever answers this question.

who needs a limit?

f = x + 4/x
f' = 1 - 4/x^2
So, f'(4) = 1 - 4/16 = 3/4

The tangent line at (4,5) is thus

y-5 = 3/4 (x-4)

using the limit

f(4+h)-f(4)
= (4+h)+4/(4+h) - (4 + 4/4)
= ((4+h)^2+4)/(4+h) - 5
= ((4+h)^2+4-5(4+h))/(4+h)
= (h^2+3h)/(4+h)
= h(h+3)/(h+4)

Now divide that by h and you get
(h+3)/(h+4)
as h->0, that -> 3/4

To find the equation of the tangent line to the graph of f at the given point (4,5), we can use calculus and the concept of the limit process.

1. First, let's find the derivative of the function f(x). The derivative gives us the slope of the tangent line at any point on the graph of f.

To find the derivative using the limit process, we start by finding the difference quotient:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

2. Substitute the given function f(x) = x + (4/x) into the difference quotient:

f'(x) = lim(h->0) [(x + h + 4/(x + h)) - (x + 4/x)] / h

3. Simplify the numerator:

f'(x) = lim(h->0) [(x(x + h) + 4) - (x(x + h) + 4)] / [h(x + h)x]

f'(x) = lim(h->0) [4/(h(x + h)x)]

4. Factor out h from the denominator:

f'(x) = lim(h->0) [4/(hx(x + h))]

5. Cancel out the h from the numerator and denominator:

f'(x) = lim(h->0) [4/(x(x + h))]

6. Now we can evaluate the limit:

f'(x) = 4/(x(x)) = 4/x^2

7. Now we have the derivative of f(x) as f'(x) = 4/x^2.

To find the equation of the tangent line at the point (4,5), we need to find the slope and the y-intercept of the line.

1. Substitute the x-coordinate of the given point (4,5) into the derivative f'(x) = 4/x^2:

f'(4) = 4/4^2 = 4/16 = 1/4

2. The slope of the tangent line is 1/4.

3. We can use the point-slope form of a line to find the equation. The point-slope form is:

y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.

Substituting the values (4,5) and m=1/4:

y - 5 = (1/4)(x - 4)

4. Simplify and rewrite the equation in the slope-intercept form:

y - 5 = (1/4)x - 1

y = (1/4)x + 4

Therefore, the equation of the tangent line to the graph of f(x) = x + (4/x) at the point (4,5) is y = (1/4)x + 4.