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)?
"How do you" is given in the explanation.
Given:
The limit definition is
Substitute these functions into the definition:
Remove the canceled factors:
Distribute the minus sign in the numerator through the parenthesis:
More canceling:
Remove the cancelled terms:
Now, it is ok to let the limit go to zero:
Remove the 0:
The numerator becomes a square:
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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:

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

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).

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

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