Q. 4 - Calculus

Question

Explain graphically why it makes sense that a limit $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ of the indeterminate form $\frac{\infty }{\infty }$ would be related to $\underset{x\to \infty }{\lim }\frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}$.

Step-by-Step Solution

Answer

Limits of the indeterminate form $\frac{\infty }{\infty }$ are related to the limit of the derivatives of its numerator and denominator functions.

Step 1. Given information

We have to prove graphically that a limit $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ of the indeterminate form $\frac{\infty }{\infty }$ would be related to $\underset{x\to \infty }{\lim }\frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}$.

Step 2. Proof of the given question graphically.

Let us take an example

$\underset{x\to \infty }{\lim }\frac{1-2x^2}{2-3x^2}$

The value of the limit is in the form of $\frac{\infty }{\infty }$.

The graph of $f\left(x\right)=1-2x^2$ and $g\left(x\right)=2-3x^2$ are

As $x\to \infty$, the graph of both the functions become vertical same as the vertical tangent at $x=\infty$. So,  $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ of the indeterminate form $\frac{\infty }{\infty }$ would be related to $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$.