How do you use the limit definition to find the slope of the tangent line to the graph #f(x)=x^3#?
Expand, reduce and evaluate the limit.
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To use the limit definition to find the slope of the tangent line to the graph of f(x) = x^3, we can follow these steps:

Start with the equation of the function: f(x) = x^3.

Choose a point on the graph of the function, let's say (a, f(a)).

Select a second point on the graph that is very close to the first point, let's say (a + h, f(a + h)).

Calculate the slope of the secant line passing through these two points using the formula: (f(a + h)  f(a)) / (a + h  a).

Simplify the expression obtained in step 4.

Take the limit as h approaches 0 of the expression obtained in step 5.

The result of step 6 will give us the slope of the tangent line to the graph of f(x) = x^3 at the point (a, f(a)).
Therefore, by following these steps, we can use the limit definition to find the slope of the tangent line to the graph of f(x) = x^3.
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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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