Laws of Combining Limits

0.1 Laws of Combining Limits Just as polynomials are linear combinations of power functions, most functions can be written as a combination of simpler functions. For instance, the function f (x) = 2 · x + 5 can be viewed as a combination of the functions g(x) = 2, h(x) = x, and i(x) = 5, wherein f (x) = g(x) · h(x) + i(x). Ideally, we should be able to infer something about a limit involving f using information of limits involving g, h, and i. If we are able to find rules for combining limits of functions, then we can replace the task of proving very complicated limits with that of using knowledge of basic functions, and the corresponding laws for combining limits. For functions we can construct from functions with known limits, this will allow us to avoid working with the ε-δ definition of the limit directly, which is often extremely difficult. As we increase the library of functions for which we know their limits, the more limits we will be able to evaluate while avoiding formal proof. We begin by establishing limits for two very basic functions. Claim 0.1.1. lim c = c x→x0 Proof. Fix ε > 0. For any x ∈ R, we have f (x) = c. Therefore, for any δ > 0, and x with |x − x0 | < δ we have |f (x) − c| = |c − c| = 0 < ε. Claim 0.1.2. lim x = x0 x→x0 Proof. Fix ε > 0. Set δ = ε. For any x with |x − x0 | < δ we have |f (x) − x0 | = |x − x0 | < δ = ε. Now that we’ve proven limits for the most basic building blocks, we need to look at rules for combining them. Note that these results follow from the way in which we have defined the limit; thus, they are inescapable. By considering this notion of the limit, it must have the following properties. Theorem 0.1.3 (Rules for Combining Limits). Suppose limx→x0 f (x) = L and limx→x0 g(x) = M where L, M ∈ R. The following results are true. 1. Constant Multiple Rule: limx→x0 k · f (x) = k · L, 2. Sum Rule: limx→x0 [f (x) + g(x)] = L + M 3. Product Rule: limx→x0 [f (x) · g(x)] = L · M k∈R 4. Quotient Rule: limx→x0 f (x) g(x) = L M, M 6= 0 The proofs of the first of these two results are well within our grasp. For the first we can fix an ε and choose δ so that |f (x) − L| < ε/|k|. For the second we just need to choose a δ so that |f (x) − L| < ε/2 and |g(x) − M | < ε/2. Once we’ve done so we simply apply the triangle inequality. The proof of the product rule requires the notion of boundnedness (or local boundedness more precisely), which we are not yet ready to tackle. Finally, the quotient rule requires assurity that our function stays sufficiently far away from 0 (which it must for the limit to exist as nonzero), but we will also avoid that proof at the moment. Using these laws in conjunction with the limits of f (x) = x and f (x) = c at any point we are now equipped to deal with the limit of any polynomial at any point. Example 1. Find lim (3x2 + x − 4) x→1 Solution First we apply the sum rule to spread the limit to each of the individual terms, and then we can evaluate limit of the second two terms immediately. For the first, we use the constant product rule to separate 3 from x2 , and finally the product rule (using x · x) to evaluate the limit for x2 . Thus,     lim (3x2 + x − 4) = 3 lim x + lim x · lim x + lim (−4) = 3 · 1 + 1 · 1 − 4 = −2 x→1 x→1 x→1 x→1 x→1 Example 2. Find limx→x0 f (x) for all x0 ∈ [0, 4], for f defined below. If the limit does not exist at any x0 , state why.  x 0≤x<1     0 x=1    1 1