Question

How do I find the limit as x approaches infinity of $\dfrac{{{x}^{3}}}{3x}$ ?

Answer 1

In any x tends to infinity the value of $\dfrac{f\left( x \right)}{g\left( x \right)}$ depends up on the highest power of x in f( x ) and highest power of g( x ) where f and g are algebraic function of x. If f has greater higher power than g, then the value of $\dfrac{f\left( x \right)}{g\left( x \right)}$ will tend to infinity as x tends to infinity. If g has greater highest power then the value of $\dfrac{f\left( x \right)}{g\left( x \right)}$ will tend to 0 as x tends to infinity. If both has same highest power then the value of $\dfrac{f\left( x \right)}{g\left( x \right)}$ will tend to $\dfrac{\text{coefficient of highest power of x in f}}{\text{coefficient of highest power of x in g}}$ as x tends to infinity.

Complete step by step solution:
We have the value of $\displaystyle \lim_{x \to \infty }\dfrac{{{x}^{3}}}{3x}$
Highest power of ${{x}^{3}}$ is equal to 3 and the highest power of 3x is equal to 1. So the highest power of ${{x}^{3}}$ is greater than the highest power of 3x .
We know that if the highest power of x in the numerator is greater than the highest power of x in the denominator, then the value of the function will tend to infinity.
So the value of $\displaystyle \lim_{x \to \infty }\dfrac{{{x}^{3}}}{3x}$ will tend to infinity as x tend to infinity.
We can solve it by another method by, we can write $\dfrac{{{x}^{3}}}{3x}$ as $\dfrac{{{x}^{2}}}{3}$
So we can write $\displaystyle \lim_{x \to \infty }\dfrac{{{x}^{3}}}{3x}$ is equal to $\displaystyle \lim_{x \to \infty }\dfrac{{{x}^{2}}}{3}$ .
$\dfrac{{{x}^{2}}}{3}$ will tend to infinity as x tends to infinity .
So $\displaystyle \lim_{x \to \infty }\dfrac{{{x}^{3}}}{3x}$ tends to infinity.

Note: In the problem $\displaystyle \lim_{x \to 0}\dfrac{f\left( x \right)}{g\left( x \right)}$ the above rule is reversed. If function f has greater highest power then $\displaystyle \lim_{x \to 0}\dfrac{f\left( x \right)}{g\left( x \right)}$ will be equal to 0 . If function g has greater highest power than function f then $\displaystyle \lim_{x \to 0}\dfrac{f\left( x \right)}{g\left( x \right)}$ will tend to infinity . If both has same highest power then $\displaystyle \lim_{x \to 0}\dfrac{f\left( x \right)}{g\left( x \right)}$ is equal to $\dfrac{\text{constant}\text{ present}\text{ in}\text{ function}\text{ f}}{\text{constant}\text{ present}\text{ in}\text{ function}\text{ g}}$ .