Lecture 6: Indeterminate forms

1. Indeterminate form

When calculating the limits of functions, we sometimes need to deal with limits of the following form: f(x) lim x→a g(x) where both f(x) and g(x) approaches zero, or both approaches ∞ when x tends to a. sin x log(1 + x) x For example lim , lim , lim are limits of such type. x→0 x x→0 x x→∞ ex

Indeterminate form: Let a be a real number, ∞ or −∞. The limit

f(x) lim x→a g(x)

is called an indeterminate form of • 0 0 type if both f(x) and g(x) approaches zero when x tends to a; • ∞ ∞ ∞ type if both f(x) and g(x) approaches when x tends to a;

The definition applies to left limit or right limit as well. For some indeterminate forms, we can show its limit using known techniques. For sin x x instance, we have proved that lim = 1, lim = 0. When the function gets x→0 x x→∞ ex complicated, it is usually hard to find the limit by using old methods.

Question: What are the limits of the following functions? log(1 + x) (1). lim . x→0+ x x − sin x (2). lim . x→0 x log(1 + x) − x2 To answer the question, we need to introduce new tools. In this lecture, we involve mainly two techniques: L’Hopital’s rule and Taylor’s formula.

1 2. L’Hopital’s rule

f(x) 0 ∞ L’Hopital’s rule: Suppose lim is an indeterminate form of type or ∞ type. Suppose x→a g(x) 0 f ′(x) f(x) and g(x) are differentiable around a. If lim = L (a real number, ∞ or −∞) x→a g′(x) exists, then we have f(x) f ′(x) lim = lim = L. x→a g(x) x→a g′(x) The rule applies to a left limit or a right limit as well.

0 In the lecture, we will give a proof of the rule for the case that the limit is a 0 type and a is a real number. The other cases can be proved using the same principle and more discussions. Example: log(1 + x) (1). lim . x→0+ x log(x2 + 3x + 5) (2). lim x→∞ cosh x 3. Using Taylor’s formula to calculate indeterminate forms

Suppose around a ∈ R, f(x) and g(x) are both Cn+1-functions. Then by Taylor’s formula, we have

f (n)(a) f(x) = f(a) + f ′(a)(x − a) + ··· + (x − a)n + o((x − a)n) n! g(n)(a) g(x) = g(a) + g′(a)(x − a) + ··· + (x − a)n + o((x − a)n). n! Therefore

′ − ··· f (n)(a) − n − n f(x) f(a) + f (a)(x a) + + n! (x a) + o((x a) ) lim = lim (n) . x→a g(x) x→a ′ − ··· g (a) − n − n g(a) + g (a)(x a) + + n! (x a) + o((x a) )

Suppose f (k)(a) is the first non-zero higher order derivative for f(x) at a. Namely f(a) = f ′(a) = ··· = f (k−1)(a) = 0 but f (k)(a) ≠ 0. Suppose g(l)(a) is the first non-zero higher order derivative for g(x) at a. Then we have

f (k)(a) − k ··· f (n)(a) − n − n f(x) k! (x a) + + n! (x a) + o((x a) ) lim = lim (l) (n) x→a g(x) x→a g (a) − l ··· g (a) − n − n l! (x a) + + n! (x a) + o((x a) )

2  f(k)(a)  f (k)(a)  k! = k = l  g(l)(a) g(l)(a) = l! . 0 k > l  ∞ k < l

Note: L’Hopital’s rule applies when a = ∞, but the method of Taylor’s formula only works for a real number a. Example: x − sin x (1). lim . x→0 x log(1 + x) − x2 4. Other indeterminate forms

There are other types of indeterminate forms: 0 1 (1). ∞ -type, such as lim x x x→∞ 1 (2). 10-type, such as lim (1 + )x x→∞ x (3). 00-type, such as lim xx x→0+ (4). ∞ − ∞-type, such as lim x − log x x→∞ (5). ∞ · 0-type, such as lim log x sin x x→0+ 0 ∞ The basic method of calculating them is to transform each of them into a 0 or ∞ -type. Example: 1 1 (1). lim − . x→0+ x sin x 1 (2). lim (1 − )x x→∞ x 5. Homework

Q.1 Calculate the following limits. x2 (1). lim x→−∞ e1−x log(1 + x) sin x − x2 (2). lim x→0 x2ex + 2(cos x − 1)

1 (3). lim x x x→∞

3 Q.2 Use Taylor’s theorem to solve the question. Suppose f(x) is a C∞-function around zero and log(1 + x) − f(x) lim = 0. x→0 x2 (1). Calculate f(0), f ′(0) and f ′′(0). f(x) (2). Calculate lim . x→0 x

Q.3 Function f(x):(−1, 1) → R is defined as follows. { − − log(1 x)(1 cos x) , x ≠ 0 f(x) = x sin x 0, x = 0.

(1). Show that f(x) is continuous at zero.

(2). Use the definition of derivative to calculate f ′(0), if it exists.

Homework Submission Guidelines:

(1). Use A4 size papers

(2). Make sure your full name, your ID appears at the top of the first page.

(3). Bind them if your homework is written in multiple papers.

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