Problem
5 2 nd understand (R\&U): Stewart's pp 95-113 The given limit is a derivative of a function $f(x)$ at point $a$. Find the expression that represents the function and the value of $a$. \[ \lim _{h \rightarrow 0} \frac{1-(1+h)^{27}}{h} \] Note: Due to the choice of the formula for the derivative, the limits do not depend on the value of $m$. Hence the limit for the functions $f(x)$ and $f(x+m)$, where $m$ is any real number, are the same in this case. Give the answer for $m=0$. \[ \begin{array}{l} f(x)=\square+C, C=\mathbb{R} \\ a=1 \quad \text { Clear all } \end{array} \] RADE ANSWER
Solution
Step 1 ：The given limit is the definition of the derivative of a function at a point.
Step 2 ：The function is \(f(x) = x^{27}\) and the point is \(a = 1\).
Step 3 ：The derivative of \(f(x)\) is \(f'(x) = 27x^{26}\), so the limit is equivalent to \(f'(1) = 27*1^{26} = 27\).
Step 4 ：The expression that represents the function is \(f(x) = x^{27} + C\), where \(C\) is any real number, and the value of \(a\) is 1.
Step 5 ：\(\boxed{f(x)=x^{27}+C, C=\mathbb{R}}\)
Step 6 ：\(\boxed{a=1}\)
