Use the definitions given in Exercise 57 to prove the following infinite limits.


lim x→1^- 1 / 1 − x=∞

Verified step by step guidance

1. Start by recalling the definition of an infinite limit: lim x→c f(x) = ∞ means that for every M > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then f(x) > M.
2. In this case, we need to analyze the function f(x) = 1 / (1 - x) as x approaches 1 from the left (x → 1^-).
3. Observe that as x approaches 1 from the left, the expression 1 - x approaches 0 and is positive, which means that f(x) = 1 / (1 - x) will increase without bound.
4. To formalize this, choose an arbitrary positive number M and set up the inequality 1 / (1 - x) > M, which simplifies to 1 > M(1 - x).
5. Solve for x to find the corresponding δ: rearranging gives x > 1 - 1/M, and thus, we can choose δ = 1 - 1/M to ensure that |x - 1| < δ leads to f(x) > M.