How do you use the limit definition to find the slope of the tangent line to the graph #f(x) = sqrt(x+1)# at (8,3)?

Question

Evelyn Agard · Accepted Answer

see below: #1/6# #f'(x) = (sqrt{x + h + 1} - sqrt{x + 1})/ h ]_{h o 0}# # = ( (sqrt{x + h + 1} + sqrt{x + 1}) (sqrt{x + h + 1} - sqrt{x + 1} ) )/ ((sqrt{x + h + 1} + sqrt{x + 1}) h) ]_{h o 0} # # = ( x + h + 1 - (x + 1)) / ((sqrt{x + h + 1} + sqrt{x + 1}) h) ]_{h o 0} # # = ( h) / ((sqrt{x + h + 1} + sqrt{x + 1}) h) ]_{h o 0} # # = ( 1) / ((sqrt{x + h + 1} + sqrt{x + 1}) ) ]_{h o 0} # now #sqrt{x + h + 1) ]_{h o 0} = sqrt{x + 1)# So the limit is: # ( 1) / (2sqrt{x + 1} ) # # ( 1) / (2sqrt{8 + 1} ) = 1/6#

Emily Ammerman · Answer

undefined To find the slope of the tangent line to the graph of f(x) = sqrt(x+1) at the point (8,3) using the limit definition, we can follow these steps:

1. Start with the equation of the function: f(x) = sqrt(x+1).

2. Determine the derivative of the function f(x) using the limit definition. The derivative, denoted as f'(x), represents the slope of the tangent line at any given point.

3. Apply the limit definition of the derivative to find f'(x). The limit definition states that f'(x) = lim(h→0) [f(x+h) - f(x)] / h.

4. Substitute the given point (8,3) into the equation f(x) = sqrt(x+1) to find the value of f(8).

5. Plug the values of f(x+h), f(x), and h into the limit definition equation, using f(8) as f(x) and h as the variable.

6. Simplify the equation and evaluate the limit as h approaches 0.

7. The resulting value will be the slope of the tangent line to the graph of f(x) = sqrt(x+1) at the point (8,3).