How do you use the limit definition to find the slope of the tangent line to the graph #f(x)= 1/(x+2)# at (0,1/2)?

Answer 1

"How do you" is given in the explanation.

Given:

#f(x) = 1/(x + 2)#

The limit definition is

#lim_(hto0){f(x + h) - f(x)}/h#
To create #f(x + h)#, you merely substitute #x + h# for every #x#:
#f(x + h) = 1/(x + h + 2)#

Substitute these functions into the definition:

#lim_(hto0){1/(x + h + 2) - 1/(x + 2)}/h#
Multiply by 1 in the form of #{(x + h + 2)(x + 2)}/{(x + h + 2)(x + 2)}#:
#lim_(hto0){1/(x + h + 2) - 1/(x + 2)}/h{(x + h + 2)(x + 2)}/{(x + h + 2)(x + 2)}#
Multiply the numerators on the top and the #h# on the bottom:
#lim_(hto0){((x + h + 2)(x + 2))/(x + h + 2) - ((x + h + 2)(x + 2))/(x + 2)}/(h(x + h + 2)(x + 2))#
What this does is that makes the first term in the numerator become #(x + 2)# by cancelling the denominator, #(x + h + 2)#, and it makes the second term in the numerator become #x + h + 2# by canceling the denominator, #x + 2#.
#lim_(hto0){(cancel(x + h + 2)(x + 2))/cancel(x + h + 2) - ((x + h + 2)cancel(x + 2))/cancel(x + 2)}/(h(x + h + 2)(x + 2))#

Remove the canceled factors:

#lim_(hto0){(x + 2) - (x + h + 2)}/(h(x + h + 2)(x + 2))#

Distribute the minus sign in the numerator through the parenthesis:

#lim_(hto0){x + 2 -x - h - 2}/(h(x + h + 2)(x + 2))#

More canceling:

#lim_(hto0){cancel(x) cancel(+ 2) cancel(-x) - h cancel(- 2)}/(h(x + h + 2)(x + 2))#

Remove the cancelled terms:

#lim_(hto0){ - h}/(h(x + h + 2)(x + 2))#
#-h/h# becomes -1:
#lim_(hto0){ -1}/((x + h + 2)(x + 2))#

Now, it is ok to let the limit go to zero:

#{ -1}/((x + 0 + 2)(x + 2))#

Remove the 0:

#{ -1}/((x + 2)(x + 2))#

The numerator becomes a square:

#{ -1}/((x + 2)^2)#
Substitute the x coordinate for the point #(0, 1/2)# to obtain the slope, m, of the tangent line:
#m = { -1}/((0 + 2)^2)#
#m = -1/4#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the slope of the tangent line to the graph of f(x) = 1/(x+2) at (0,1/2) using the limit definition, we can follow these steps:

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

  2. Determine the derivative of the function f(x) using the limit definition of the derivative. The derivative of f(x) is given by the formula: f'(x) = lim(h→0) [f(x+h) - f(x)] / h.

  3. Substitute the given point (0,1/2) into the derivative equation to find the slope of the tangent line at that point. Plug in x = 0 and solve for f'(0).

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

  5. The resulting value will be the slope of the tangent line to the graph of f(x) at the point (0,1/2).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7