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I'm new to calculus and self-teaching so please be kind to me :)

The book I am reading says I should convince myself that the limit of cos(x)/x as x->0 does not exist so i thought i'd check my thinking on here.

by very small i mean close to 0

The lim as x->0+ of cos(x)/x:

As x gets very small cos(x) approaches 1

As x gets very small x becomes a small positive number

So 1/tiny positive number = + infinity

The lim as x->0- of cos(x)/x:

As -x gets very small cos(x) approaches 1

As -x gets very small, x becomes a small positive number

So 1/tiny negative number = - infinity

So the left and right handed limits aren't the same so the limit does not exist.

If anyone was willing to show me how you write this properly it would be much appreciated

thanks