Question

Let f be a function defined on some interval $(a,\infty )$. Then

$\underset{x\to \infty }{\lim }f\left(x\right)=L$

Then means that the values of $f\left(x\right)$can be made arbitrarily close to _____ by taking _____ sufficiently large. In this case the line $y=L$is called a ___ _____ of the function $y=f\left(x\right)$is called a. For example, $\underset{x\to \infty }{\lim }\frac{1}{x}=$_____, and the line y=_____ is a horizontal asymptote.

Short answer

The value of limit of the given function is $L,x,\text{limit},0,0.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{x\to \infty }{\lim }f\left(x\right)=L$

Step 2. Concept  used.

Here we use $\underset{x\to \infty }{\lim }\frac{1}{x}=0$

Here we use the concept of horizontal asymptote.

Step 3. Simplification.

Given,$\underset{x\to \infty }{\lim }f\left(x\right)=L$

In general, we use the notation:$\underset{x\to \infty }{\lim }f\left(x\right)=L$

To indicate that the values of $f\left(x\right)$ become closer and closer to

$L$as $x$becomes larger and larger.

Let $f$be a function defined on some interval $(a,\infty )$. Then

$\underset{x\to \infty }{\lim }f\left(x\right)=L$

This means that the values of $f\left(x\right)$can be made arbitrarily close to $L$by taking $x$sufficiently large.

According to horizontal asymptote:

The line $y=L$is called a horizontal asymptote of the curve $y=f\left(x\right)$if:

$\underset{x\to \infty }{\lim }f\left(x\right)=L$

Therefore, our final answer in given gaps are :$L,x,\lim it,0,0$