If \(f\) is a sequence, then to say that as \(n \to \infty\), \(f(n) \to L\) means that for every \(\epsilon > 0\) there is some number \(N\) such that if \(n\) is a positive integer and \(n > N\), then \(|f(n) - L| < \epsilon\). We defined \(\lim_{n \to \infty} f(n)\) to be the unique \(L\) such that as \(n \to \infty\), \(f(n) \to L\), if there is such an L.

If \(\lim_{x \to \infty} = L\), then \(\lim_{n \to \infty} = L\), where in the second limit \(n\) stands for an integer.

If \(\lim_{n \to \infty} f(n) = L\), then as we go through the list f(1), f(2), f(3), …., the numbers in the list eventually get close to \(L\). In this situation, we say that the sequence \((f(n))^{\infty}_{n=1}\) converges to \(L\).

If \(\lim_{n \to \infty} f(n)\) is undefined, then we say that the sequence \((f(n))^{\infty}_{n=1}\) diverges.

Suppose that \((u_n)^{\infty}_{n=1}\) and \((v_n)^{\infty}_{n=1}\) are two sequences of numbers such that \(u_1 \leq u_2 \leq u_3 \leq ....\cdot\) and \(v_1 \geq v_2 \geq v_3 \geq ....\cdot\) Suppose also that for every \(n\), \(u_n \leq v_n\), and \(\lim_{n \to \infty} (v_n - u_n) = 0\). Then there is a unique number \(c\) such that for every \(n\), \(u_n \leq c \leq v_n\). Furthermore, \(\lim_{n \to \infty} u_n = \lim_{n \to \infty} v_n = c\); in fact, since for every \(n\) we have \(u_n \leq c \leq v_n\), we can say that as \(n \to \infty\), \(u_n \to c^{\leq}\) and \(v_n \to c^{\geq}\).

Suppose that \(\lim_{x \to a}f(x) = L\) and \(r\) is any real number.

• If \(L > r\), then there is some number \(\delta > 0\) such that if \(0 < |x-a| < \delta\) then \(f(x) > r\)
• If \(L < r\), then there is some number \(\delta > 0\) such that if \(0 < |x-a| < \delta\) then \(f(x) < r\)

Suppose that \(\lim_{x \to a}f(x) = L\) and \(r\) is any real number and \(d > 0\).

• Suppose that for all \(x\), if \(0 < |x-a| < d\) then \(f(x) \leq r\). Then, \(L \leq r\)
• Suppose that for all \(x\), if \(0 < |x-a| < d\) then \(f(x) \geq r\). Then, \(L \geq r\)

This corollary states that weak inequality are presevered by limits. i.e If \(f(x) \leq r\) for \(x\) close to \(a\) but not equal to \(a\), then \(\lim_{x \to a}f(x) \leq r\).