Question

How do you find the limit of `1/(x-1)` as x approaches 1?

Answer 1

See the explanation.

Explanation:

It is relevant for the limit from which side we approach to specific point; in the other words we have to solve two limits:

Let `epsilon in R^+, epsilon->0`, then:

Form the left:
`lim_(x->1-epsilon) 1/(x-1) = lim_(epsilon->0) 1/(1-epsilon-1) = lim_(epsilon->0) 1/-epsilon = -lim_(epsilon->0) 1/epsilon = -oo`

Form the right:
`lim_(x->1+epsilon) 1/(x-1) = lim_(epsilon->0) 1/(1+epsilon-1) = lim_(epsilon->0) 1/epsilon = +oo`