How do you use the limit definition to find the slope of the tangent line to the graph #f(x)=sqrt(x+2)# at x=2?

Question

How do you use the limit definition to find the slope of the tangent line to the graph  #f(x)=sqrt(x+2)# at x=2?

Luke Alvarez · Accepted Answer

#\frac{1}{4}#

By definition, the slope of the tangent line to the graph of a given function #f(x)# at a given point #x_0# is the derivative #f'(x_0)#, which in turn is defined as follows:

#f'(x_0) = \lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}#

In your case, #f(x) = sqrt(x+2)# and #x_0 = 2#. The definition becomes

#f'(2) = \lim_{h \to 0} \frac{sqrt((2+h)+2)-sqrt(2+2)}{h} = \lim_{h \to 0} \frac{sqrt(4+h)-2}{h}#

To solve this limit we may "razionalize" the expression:

#\frac{sqrt(4+h)-2}{h} = \frac{sqrt(4+h)-2}{h} \cdot \frac{sqrt(4+h)+2}{sqrt(4+h)+2} = \frac{(4+h)-4}{sqrt(4+h)+2} = \frac{h}{sqrt(4+h)+2}#

The limit becomes

#f'(2) = \lim_{h \to 0}\frac{h}{h(sqrt(4+h)+2)} = \lim_{h \to 0}\frac{1}{sqrt(4+h)+2} = \frac{1}{sqrt(4+0)+2} = \frac{1}{2+2} = \frac{1}{4}#

Luke Alvarez · Answer

undefined To find the slope of the tangent line to the graph of f(x) = √(x+2) at x=2 using the limit definition, we can follow these steps:

1. Start with the equation of the function: f(x) = √(x+2).

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 value of x=2 into the limit definition equation: f'(2) = lim(h→0) [f(2+h) - f(2)] / h.

5. Evaluate the limit as h approaches 0. This involves plugging in the value of h=0 into the equation obtained in step 4 and simplifying the expression.

6. Simplify the expression and calculate the limit to find the slope of the tangent line at x=2.

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+2) at x=2.