Question

How do you evaluate the limit $\dfrac{1}{{{x^2}}}$ as x approaches 0?

Answer 1

Hint: In this question, we need to find the limit of a given function. In the beginning, we take different values of x, and see the nature of the function $\dfrac{1}{{{x^2}}}$ for those values. To understand this, we take smaller values of x and then square it. After that we see that the reciprocal of this ${x^2}$. We see that these values will be greater and so will be as compared to x. So, then we apply the limit as x approaches zero from the negative side and positive side and solve the given problem.

Complete step by step solution:
Given an expression of the form $\dfrac{1}{{{x^2}}}$ …… (1)
We have to find the limit of this function as x approaches zero.
Firstly, let us understand the definition of limit.
A limit is defined as a function that has some value that approaches the input.
A limit of a function is represented as : $\mathop {\lim }\limits_{x \to n} f(x) = L$,
Here $\lim $ refers to limit, it generally describes that the real valued function $f(x)$ tends to attain the limit L as x tends to n and is denoted by an arrow.
We can read this as `the limit of any given function $f$ of x as x approaches to n is equal to L.
Now consider the given problem given in the equation (1).
We give different smaller values of x and see the nature of the function $\dfrac{1}{{{x^2}}}$.
To do this we set different values of x and then square it, after that we find the reciprocal of the square of x.
For $x = 1$, we have,
${x^2} = {1^2}$
$ \Rightarrow {x^2} = 1$
$ \Rightarrow \dfrac{1}{{{x^2}}} = \dfrac{1}{1}$
$ \Rightarrow \dfrac{1}{{{x^2}}} = 1$.
For $x = \dfrac{1}{2}$, we have,
${x^2} = {\left( {\dfrac{1}{2}} \right)^2}$
$ \Rightarrow {x^2} = \dfrac{1}{4}$
$ \Rightarrow \dfrac{1}{{{x^2}}} = \dfrac{{\dfrac{1}{1}}}{4}$
$ \Rightarrow \dfrac{1}{{{x^2}}} = 4$.
For $x = \dfrac{1}{{10}}$, we have,
${x^2} = {\left( {\dfrac{1}{{10}}} \right)^2}$
$ \Rightarrow {x^2} = \dfrac{1}{{100}}$
$ \Rightarrow \dfrac{1}{{{x^2}}} = \dfrac{{\dfrac{1}{1}}}{{100}}$
$ \Rightarrow \dfrac{1}{{{x^2}}} = 100$.
From above, we see that as x gets smaller value, ${x^2}$ gets smaller value than that and , hence $\dfrac{1}{{{x^2}}}$ will be greater.
This means that the closer x goes to 0, the higher the function goes. In this case, it does not matter whether x approaches zero from the positive side or from the negative side, as the square makes it all positive.
We see that by choosing smaller and smaller values of x, the function can reach any size we want.
Mathematically it is expressed as,
$\mathop {\lim }\limits_{x \to {0^ + }} \dfrac{1}{{{x^2}}} = \mathop {\lim }\limits_{x \to {0^ - }} \dfrac{1}{{{x^2}}} = \mathop {\lim }\limits_{x \to 0} \dfrac{1}{{{x^2}}} = \infty $

Hence the limit of the function $\dfrac{1}{{{x^2}}}$ as x approaches 0 is given by infinity($\infty $).

Note: Students must know the definition of the limit of a function and how to apply the definition for a given problem. Before solving the problem, we need to see the nature of the function for different values of x. Then we need to apply the limit as x approaches from the positive side and from the negative side. If both of the values come to be the same, then it is the limit of a given function which is our required solution.