# How do you find the equation of the tangent line to the curve #y= (x-3) / (x-4)# at (5,2)?

Refer below for explanation.

We can now proceed to step 2.

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To find the equation of the tangent line to the curve y = (x-3)/(x-4) at (5,2), we need to find the slope of the tangent line at that point and then use the point-slope form of a line to write the equation.

To find the slope of the tangent line, we can use the derivative of the function. Taking the derivative of y = (x-3)/(x-4) with respect to x, we get:

dy/dx = [(x-4)(1) - (x-3)(1)] / (x-4)^2

Simplifying this expression, we have:

dy/dx = -1 / (x-4)^2

Now, we can substitute x = 5 into the derivative to find the slope at the point (5,2):

dy/dx = -1 / (5-4)^2 = -1

So, the slope of the tangent line at (5,2) is -1.

Using the point-slope form of a line, we can write the equation of the tangent line:

y - y1 = m(x - x1)

Substituting the values of (x1, y1) = (5,2) and m = -1, we have:

y - 2 = -1(x - 5)

Simplifying this equation, we get:

y - 2 = -x + 5

Rearranging the terms, we have:

y = -x + 7

Therefore, the equation of the tangent line to the curve y = (x-3)/(x-4) at (5,2) is y = -x + 7.

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To find the equation of the tangent line to the curve ( y = \frac{x-3}{x-4} ) at the point ( (5,2) ), follow these steps:

- Find the derivative of the function ( y = \frac{x-3}{x-4} ) using the quotient rule.
- Evaluate the derivative at the point ( x = 5 ) to find the slope of the tangent line.
- Use the point-slope form of a linear equation to write the equation of the tangent line using the point ( (5,2) ) and the slope found in step 2.

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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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