Question

The limit of $ \sum\limits^{1000}_{ n-1} (-1)^n \, x^n $ as $x?8$

• A. does not exist
• B. exists and equals to 0
• C. exists and approaches to $+\propto$
• D. exists and approaches $-8$

Answer 1

The Correct Option is C

\(\displaystyle\lim _{x \rightarrow \infty} \sum_{n=1}^{1000}(-1)^{n} x^{n}\)
\(=\displaystyle\lim _{x \rightarrow \infty}\left\{-x+x^{2}-x^{3}+x^{4}+\ldots+x^{1000}\right\}\)
\(=\displaystyle\lim _{x \rightarrow \infty}(-x) \cdot\left\{\frac{(-x)^{1000}-1}{(-x-1)}\right\}=\displaystyle\lim _{x \rightarrow \infty} \frac{x^{1001}-x}{x+1}\)
\(=\displaystyle\lim _{x \rightarrow \infty} \frac{x^{1000}-1}{1+\left(\frac{1}{x}\right)}=+\infty\)

When the input reaches a particular value, a function or sequence hits its limit. All significant notions in calculus and mathematical analysis, such as continuity, derivatives, and integrals, are defined in terms of limits. Let's examine the meaning and illustration of function limitations as well as some guidelines, characteristics, and limit instances.

Mathematical limits are unique real numbers. Consider the limit of a real-valued function "f" and a real number "c," which is generally defined as 

lim_x→c f(x)=L

It says, “The limit of f of x as x approaches c equals L.” 

The "lim" denotes the limit, and the right arrow denotes the fact that function f(x) approaches the limit L as x approaches c.