Limits at Infinity – Definition and Examples

Here you will learn how to solve or evaluate limits at infinity with examples.

Let’s begin –

Limits at Infinity

Algorithm to evaluate limits at infinity :

1). Write down the given expression in the form of a rational function i.e. $f(x)\over g(x)$, if it is not so.

2). If k is the highest power of x in numerator and denominator both, then divide each term in numerator and denominator by $x^k$.

3). Use the results $\displaystyle{\lim_{x \to \infty}}$ $c\over x^n$ = 0 and $\displaystyle{\lim_{x \to \infty}}$ c = c, where n > 0.

Also Read : How to Solve Indeterminate Forms of Limits

Following examples will illustrate the above algorithm.

Example : Evaluate $\displaystyle{\lim_{x \to \infty}}$ $ax^2 + bx + c\over dx^2 + ex + f$.

Solution : Here the expression assumes the form $\infty\over \infty$.

We notice that the highest power of x in both the numerator and denominator is 2.

So we divide each term in both the numerator and denominator by $x^2$.

$\therefore$  $\displaystyle{\lim_{x \to \infty}}$ $ax^2 + bx + c\over dx^2 + ex + f$

= $\displaystyle{\lim_{x \to \infty}}$ $a + {b\over x} + {c\over x^2}\over d + {e\over x} + {f\over x^2}$

= $a + 0 + 0\over d + 0 + 0$ = $a\over d$

Example : Evaluate the limit : $\displaystyle{\lim_{x \to \infty}}$ $x^2 + x + 1\over {3x^2 + 2x – 5}$.

Solution : Here the expression assumes the form $\infty\over \infty$.

We notice that the highest power of x in both the numerator and denominator is 2.

So we divide each term in both the numerator and denominator by $x^2$.

$\therefore$ $\displaystyle{\lim_{x \to \infty}}$$x^2 + x + 1\over {3x^2 + 2x – 5}$

Limit = $\displaystyle{\lim_{x \to 0}}$ $1 + x + x^2\over {3 + 2x – 5x^2}$ = $1\over 3$