How do you use the limit definition to find the slope of the tangent line to the graph #f(x)=sqrt(x)# at x=4?

Question

How do you use the limit definition to find the slope of the tangent line to the graph  #f(x)=sqrt(x)# at x=4?

Christopher Agresta · Accepted Answer

Details depend on exactly which limit definition of the slope of the tangent line you are using.

#lim_(trarr4) (f(t)-f(4))/(t-4) = lim_(trarr4)(sqrtt-sqrt4)/(t-4)# #" "# (form #0/0#) # = lim_(trarr4)((sqrtt-sqrt4))/((t-4))* ((sqrtt+sqrt4))/ ((sqrtt+sqrt4))# # = lim_(trarr4)(t-4)/((t-4)(sqrtt+sqrt4))# # = lim_(trarr4) 1/ (sqrtt+sqrt4)# # = 1/(sqrt4 + sqrt4) = 1/(2+2) = 1/4#

Isabella Ackerman · Answer

undefined To find the slope of the tangent line to the graph of f(x) = √(x) at x = 4 using the limit definition, we can follow these steps:

1. Start with the equation of the function: f(x) = √(x).
2. Determine the equation of the tangent line by finding its slope.
3. Use the limit definition of the derivative to find the slope of the tangent line.
4. Apply the limit definition by taking the derivative of the function f(x) = √(x).
5. Simplify the derivative expression.
6. Substitute the value x = 4 into the derivative expression to find the slope at x = 4.
7. Evaluate the limit as x approaches 4 to find the slope of the tangent line.

By following these steps, you can use the limit definition to find the slope of the tangent line to the graph of f(x) = √(x) at x = 4.