# Limit of the smallest function value

• MHB
• lfdahl
In summary, the limit of the smallest function value refers to the smallest possible output value that a given function can approach as the input value approaches a certain value. It can be calculated using various methods, such as algebraic manipulation, graphing, or using calculus techniques. This limit can provide information about the behavior of the function near a specific point and can be negative. It is also used in real-life applications in different fields, such as physics, economics, and computer science, to predict outcomes of systems.
lfdahl
Gold Member
MHB
Let $m_n$ be the smallest value of the function:

$$f_n(x)=\sum_{k=0}^{2n}x^k.$$

Show, that $m_n\to\frac{1}{2}$ as $n \to \infty$.

Source: Nordic Math. Contest

Suggested solution:
For $n > 1$:

$$f_n(x) = 1 + x + x^2 + …$$
$$= 1+x(1 + x^2 +x^4 + …) + x^2(1 + x^2 +x^4 + ….)$$
$= 1 + x(x+1)\sum_{k=0}^{n-1}x^{2k}$

From this we see that $f_n(x) \geq 1$, for $x \leq −1$ and $x \geq 0$. Consequently, $f_n$ attains its minimum value in the interval $(−1, 0)$. On this interval$f_n(x) = \frac{1-x^{2n+1}}{1-x}> \frac{1}{1-x} > \frac{1}{2}$So $m_n \geq \frac{1}{2}$. But$m_n \leq f_n\left ( -1 + \frac{1}{\sqrt{n}}\right ) < \frac{1}{2-\frac{1}{\sqrt{n}}}+\frac{\left ( 1-\frac{1}{\sqrt{n}} \right )^{2n+1}}{2-\frac{1}{\sqrt{n}}}$
As $n \rightarrow \infty$, the first term on the right hand side tends to the limit $\frac{1}{2}$.

In the second term, the factor$\left ( 1-\frac{1}{\sqrt{n}} \right )^{2n} = \left ( \left ( 1-\frac{1}{\sqrt{n}} \right )^{\sqrt{n}} \right )^{2\sqrt{n}}$

of the nominator tends to zero, because

$\lim_{k\rightarrow \infty }\left ( 1-\frac{1}{k} \right )^k = e^{-1} < 1$Thus,

$\lim_{n \rightarrow \infty }m_n = \frac{1}{2}.$

## What is the limit of the smallest function value?

The limit of the smallest function value refers to the smallest possible output value that a function can approach as its input approaches a certain value. It is also known as the infimum or greatest lower bound.

## Why is the limit of the smallest function value important?

The limit of the smallest function value is important because it helps us understand the behavior of a function near a certain input value. It can also help determine the existence and continuity of a function at a specific point.

## How is the limit of the smallest function value calculated?

The limit of the smallest function value is usually calculated using the concept of limits and the definition of the infimum. It involves finding the greatest lower bound of the function's output values as the input approaches a certain value.

## Can the limit of the smallest function value be negative?

Yes, the limit of the smallest function value can be negative. It can take on any real number as its value, depending on the behavior of the function near the input value.

## Is the limit of the smallest function value always defined?

No, the limit of the smallest function value is not always defined. It can be undefined if the function has a discontinuity or if the infimum of the output values does not exist as the input approaches a certain value.

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