12. Lim_(x Rarr 1)((sum_(k=1)^(100)xk))-100)/(x-1)

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Jun 26, 2024

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Evaluating the Limit: $\lim_{x\to 1} \frac{\left(\sum_{k=1}^{100} x^k\right)-100}{x-1}$

In this article, we will evaluate the limit $\lim_{x\to 1} \frac{\left(\sum_{k=1}^{100} x^k\right)-100}{x-1}$.

Step 1: Simplify the Summation

The given summation is a geometric series with first term $x$ and common ratio $x$. The formula for the sum of a geometric series is:

$\sum_{k=0}^{n-1} ar^k = a\frac{1-r^n}{1-r}$

In this case, $a=x$ and $r=x$, so we have:

$\sum_{k=1}^{100} x^k = x\frac{1-x^{100}}{1-x}$

Step 2: Simplify the Expression

Now, we can simplify the expression inside the limit:

$\frac{\left(\sum_{k=1}^{100} x^k\right)-100}{x-1} = \frac{x\frac{1-x^{100}}{1-x}-100}{x-1}$

Step 3: Evaluate the Limit

To evaluate the limit, we can start by multiplying the numerator and denominator by $1-x$ to get rid of the fraction:

$\lim_{x\to 1} \frac{x\frac{1-x^{100}}{1-x}-100}{x-1} = \lim_{x\to 1} \frac{x(1-x^{100})-100(1-x)}{(x-1)(1-x)}$

Now, we can cancel out the $(x-1)$ terms:

$\lim_{x\to 1} \frac{x(1-x^{100})-100(1-x)}{(x-1)(1-x)} = \lim_{x\to 1} \frac{x(1-x^{100})-100(1-x)}{1-x}$

Step 4: Evaluate the Limit as x→1

As $x\to 1$, the expression simplifies to:

$\lim_{x\to 1} \frac{x(1-x^{100})-100(1-x)}{1-x} = \frac{(1)(1-1^{100})-100(1-1)}{1-1}$

$= \frac{0-0}{0}$

This is an indeterminate form, but we can apply L'Hopital's rule to evaluate the limit. Taking the derivative of the numerator and denominator with respect to $x$, we get:

$\lim_{x\to 1} \frac{(-100x^{99})+100}{-1} = \frac{-100+100}{-1} = \boxed{100}$

Therefore, the evaluated limit is 100.