How do you use the limit definition to find the slope of the tangent line to the graph #f(x)=(3x-1)/(x-1)# at x=0?

Question

How do you use the limit definition to find the slope of the tangent line to the graph  #f(x)=(3x-1)/(x-1)# at x=0?

Elijah Abram · Accepted Answer

Please see below.

The slope of the line tangent to the graph of #f(x)# at #x=a# is #lim_(xrarra)(f(x)-f(a))/(x-a)#, #" "# if the limit exists. For this function, the slope of the tangent at #0# is #lim_(xrarr0)(f(x)-f(0))/(x-0) = lim_(xrarr0) ((3x-1)/(x-1)-1)/x# # = lim_(xrarr0) ((3x-1)-(x-1))/(x(x-1))# # = lim_(xrarr0) (2x)/(x(x-1))# # = lim_(xrarr0) 2/(x-1)# # = -2#

Isabella Ackerman · Answer

undefined To find the slope of the tangent line to the graph of f(x)=(3x-1)/(x-1) at x=0 using the limit definition, we can follow these steps:

1. Start with the equation of the function: f(x) = (3x-1)/(x-1).

2. Determine the equation of the tangent line. The slope of the tangent line is equal to the derivative of the function at the given point.

3. Find the derivative of f(x) using the limit definition. The derivative of f(x) is given by the limit as h approaches 0 of [f(x+h) - f(x)]/h.

4. Substitute the given value of x=0 into the derivative expression obtained in step 3.

5. Simplify the expression and evaluate the limit as h approaches 0.

6. The resulting value will be the slope of the tangent line to the graph of f(x) at x=0.