Question
4. [-/0.88 Points] DETAILS MY NOTES SCALCET9 2.7.045. The limit represents the derivative of some function \( f \) at some number \( a \). State such an \( f \) and \( a \). \[ \begin{array}{l}\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3} \\ f(x)=x^{5}, \quad a=27 \\ f(x)=x^{4}, \quad a=3 \\ f(x)=x^{4}, \quad a=81 \\ f(x)=\sqrt[4]{x}, \quad a=81 \\ f(x)=x^{4}, \quad a=4\end{array} \]
Ask by Joseph Stewart. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The limit represents the derivative of \( f(x) = x^{4} \) at \( a = 3 \).
Solution
We are given the limit
\[
\lim_{x \rightarrow 3} \frac{x^{4}-81}{x-3}.
\]
Recall that the derivative of a function \( f \) at a point \( a \) is defined as
\[
f'(a)=\lim_{x\rightarrow a} \frac{f(x)-f(a)}{x-a}.
\]
In order to match the given limit with the definition of the derivative, we compare:
\[
\frac{f(x)-f(3)}{x-3}=\frac{x^{4}-81}{x-3}.
\]
This comparison suggests that
\[
f(x)=x^{4} \quad \text{and} \quad f(3)=81.
\]
Thus, the function \( f \) is
\[
f(x)=x^{4},
\]
and the value \( a \) at which the derivative is taken is
\[
a=3.
\]
Therefore, the limit represents the derivative \( f'(3) \) where \( f(x)=x^{4} \).
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Simplify this solution Mind Expander
Consider the function \( f(x) = x^4 \) and the point \( a = 3 \). When evaluating the limit, you can observe that as \( x \) approaches 3, the expression \( \frac{x^4 - 81}{x - 3} \) simplifies nicely since \( x^4 - 81 \) can be factored into \( (x^2 - 9)(x^2 + 9) \). This denotes that the limit represents the derivative of \( f(x) \) at that point. To find the derivative using the limit, you realize that substituting 3 into the function \( f(x) = x^4 \) gives you \( f'(3) = 4(3^3) = 108 \). This shows the beauty of limits and derivatives in calculus, linking them to real-world problems like determining slopes or rates of change in various contexts.
