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 $a$ .
$lim_{h \to 0} \frac{1 - (1 + h)^{27}}{h}$
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 $m = 0$ .
$\begin{array}{l} f (x) = ◻ + C, C = R \\ a = 1 Clear all \end{array}$
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Answer

$a = 1$

Steps

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 $f^{'} (x) = 27 x^{26}$ , 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 ： $f (x) = x^{27} + C, C = R$

Step 6 ： $a = 1$