L'Hospital's Rule and Taylor Series

Section 1: L'Hospital's Rule

Theorem 1.1: L'Hospital's Rule (0/0 form)

If $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$ , and $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ exists (or is ±∞), then:

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Theorem 1.2: L'Hospital's Rule (∞/∞ form)

If $\lim_{x \to a} f(x) = \pm\infty$ and $\lim_{x \to a} g(x) = \pm\infty$ , and $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ exists (or is ±∞), then:

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Example 1.1: L'Hospital Application

Evaluate: $\lim_{x \to 0} \frac{\sin x}{x}$

Solution: This is 0/0 form. Apply L'Hospital:

\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1

Example 1.2: Multiple Applications

Evaluate: $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$

Solution: Apply L'Hospital twice:

\lim_{x \to 0} \frac{e^x - 1}{2x} = \lim_{x \to 0} \frac{e^x}{2} = \frac{1}{2}

Section 2: Taylor's Theorem

Theorem 2.1: Taylor's Theorem with Lagrange Remainder

If $f$ is $n+1$ times differentiable on an interval containing $x_0$ and $x$ , then:

f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k + R_n(x)

where the Lagrange remainder is:

R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}

for some $\xi$ between $x_0$ and $x$ .

Example 2.1: Maclaurin Series of e^x

Find: Maclaurin series of $e^x$

Solution: Since $f^{(k)}(0) = e^0 = 1$ for all $k$ :

e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

Example 2.2: Maclaurin Series of sin x

Find: Maclaurin series of $\sin x$

Solution: The derivatives cycle: sin, cos, -sin, -cos, ...

\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!}

Section 3: Standard Expansions

Common Maclaurin Series

$e^x$ :

1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

$\sin x$ :

x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

$\cos x$ :

1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

$\ln(1+x)$ :

x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots

Section 4: Applications of L'Hospital's Rule

Example 4.1: 0/0 Form

Find: $\lim_{x \to 0} \frac{\sin x}{x}$

$\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1$

Example 4.2: ∞/∞ Form

Find: $\lim_{x \to \infty} \frac{x}{e^x}$

$\lim_{x \to \infty} \frac{x}{e^x} = \lim_{x \to \infty} \frac{1}{e^x} = 0$

Section 5: Taylor Polynomials

Definition 5.1: Taylor Polynomial

The n-th degree Taylor polynomial of $f$ at $a$ is:

P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k

Example 5.1: Taylor Polynomial

The 3rd degree Taylor polynomial of $e^x$ at 0 is:

P_3(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}

Section 6: Remainder Term

Theorem 6.1: Taylor's Theorem with Remainder

If $f$ has $n+1$ derivatives on an interval containing $a$ , then:

f(x) = P_n(x) + R_n(x)

where $R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}$ for some $\xi$ between $a$ and $x$ .

Section 7: Convergence of Taylor Series

Theorem 7.1: Convergence Criterion

The Taylor series converges to $f(x)$ if and only if $\lim_{n \to \infty} R_n(x) = 0$ .

Example 7.1: Convergence

The Taylor series for $e^x$ converges for all $x$ because $R_n(x) \to 0$ as $n \to \infty$ .

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Practice Quiz: L'Hospital's Rule and Taylor Series

10

Questions

0

Correct

0%

Accuracy

1

Evaluate

\lim_{x \to 0} \frac{\sin x}{x}

using L'Hospital's rule.

Easy

Not attempted

2

Evaluate

\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}

.

Medium

Not attempted

3

What is the Maclaurin series of

e^x

up to

x^3

?

Easy

Not attempted

4

The Lagrange remainder for Taylor's theorem is:

Medium

Not attempted

5

Evaluate

\lim_{x \to 0} \frac{x - \sin x}{x^3}

using Taylor expansion.

Medium

Not attempted

6

For which form can L'Hospital's rule NOT be directly applied?

Easy

Not attempted

7

Evaluate

\lim_{x \to 0^+} x \ln x

.

Medium

Not attempted

8

Evaluate

\lim_{x \to 0} \frac{1 - \cos x}{x^2}

.

Easy

Not attempted

9

What is the coefficient of

x^4

in the Maclaurin series of

\cos x

?

Easy

Not attempted

10

Evaluate

\lim_{x \to 0} \frac{\tan x - x}{x^3}

.

Medium

Not attempted

Frequently Asked Questions

When can I use L'Hospital's rule?

L'Hospital's rule applies to 0/0 and ∞/∞ indeterminate forms. Both f and g must be differentiable near the limit point, and the limit of f'/g' must exist (or be ±∞).

What is Taylor's theorem?

Taylor's theorem states that if f is n+1 times differentiable, then f(x) = P_n(x) + R_n(x), where P_n is the n-th degree Taylor polynomial and R_n is the remainder term.

What is the difference between Taylor series and Maclaurin series?

A Maclaurin series is a special case of a Taylor series centered at x₀ = 0. Taylor series can be centered at any point x₀.

How do I find the Taylor expansion of a function?

Compute f(x₀), f'(x₀), f''(x₀), ..., f^(n)(x₀), then use the formula: f(x) = Σ [f^(k)(x₀)/k!] (x - x₀)^k.

What is the Lagrange remainder?

The Lagrange remainder is R_n(x) = [f^(n+1)(ξ)/(n+1)!] (x - x₀)^(n+1) for some ξ between x₀ and x. It provides an error bound for the Taylor approximation.