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Homework Statement
Estimate the limit by substituting smaller and smaller values of h.
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Homework Equations
The Attempt at a Solution
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The purpose of estimating a limit with smaller values of h is to get a more precise approximation of the actual limit. By using smaller values of h, we can get closer and closer to the true value of the limit, giving us a better understanding of the behavior of the function near the limit point.
The values of h should be chosen to be smaller and smaller, approaching zero. This allows us to see the behavior of the function as h gets closer and closer to zero, giving us a more accurate estimate of the limit.
If the values of h are not small enough, the estimate of the limit may be less accurate and may not reflect the true behavior of the function at the limit point. Additionally, using larger values of h may not allow us to see any potential oscillations or abrupt changes in the function near the limit point.
No, estimating a limit with smaller values of h does not guarantee an accurate result. It can give us a more precise approximation, but it is still an estimation and may not reflect the exact value of the limit. Other methods, such as algebraic manipulation or L'Hopital's rule, may be needed for a more accurate result.
Yes, there are limitations to estimating a limit with smaller values of h. Depending on the complexity of the function, using smaller values of h may be computationally intensive and time-consuming. Additionally, there may be cases where the limit cannot be accurately estimated using this method and other approaches may be necessary.