Q. 2 - Calculus

Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible. 

the intuitive meaning of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=L$

Step-by-Step Solution

Answer

 Vertical Asymptotes: If f is a function and c and L are real numbers where the value of $f\left(x\right)$reaches infinity as x reaches c then we can say that infinity is the limit of $f\left(x\right)$as x approaches c and $x=c$ is vertical asymptotes.

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

Horizontal Asymptotes: If f is a function and c and L are real numbers where the value of $f\left(x\right)$reaches L as x reaches infinity then we can say that L is the limit of $f\left(x\right)$as x approaches infinity and $y=L$ is horizontal asymptotes.

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

Step 1. Given information.

The given functions with limits are 

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

Step 2. Description of Vertical Asymptotes

Vertical Asymptotes: If f is a function and c and L are real numbers where the value of $f\left(x\right)$reaches infinity as x reaches c then we can say that infinity is the limit of $f\left(x\right)$as x approaches c and $x=c$ is vertical asymptotes.

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

For example in$\underset{x\to 0}{\lim }f\left(x\right)=\infty$, function value reaches to infinity as x reaches to zero and vertical asymptotes at $x=0$.

Step 3. Description of Horizontal Asymptotes

Horizontal Asymptotes: If f is a function and c and L are real numbers where the value of$f\left(x\right)$reaches L as x reaches infinity then we can say that L is the limit of $f\left(x\right)$as x approaches infinity and $y=L$ is horizontal asymptotes.

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

For example in $\underset{x\to \infty }{\lim }f\left(x\right)=0$, function value reaches to zero as x reaches to infinity and horizontal asymptotes at  $y=0.$