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