2WBB0Calculus I

2WBB0 · Topic 06

Taylor expansions and hard limits

Use the supplied standard series to expose cancellation and prove local approximations.

What you need to be able to do

Recall which standard Taylor series are supplied on the exam sheet.
Expand composite expressions to the first nonzero surviving order.
Control O-terms and divide only after factoring the dominant power.

1. Reading and substituting standard series

The exam sheet supplies several standard expansions around zero. The skill is not memorizing coefficients; it is substituting a new inner expression and tracking the resulting powers.

If x is replaced by x², every power doubles. Write the new O-term explicitly so you know which orders are safe.

Exam method

Choose the supplied series matching the outer function.
Substitute the complete inner expression for x.
Expand only as far as the question's denominator or cancellation requires.
Rewrite O((xᵐ)ⁿ) as O(xᵐⁿ).

Reading and substituting standard series: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Reading and substituting standard seriesSelf-review

Worked example 1

\sin(x^2)

2. Finding the first surviving order

In a Taylor limit, the numerator is designed to cancel. Expand every term far enough to find the first nonzero coefficient, not merely to the first familiar term.

The denominator tells you the minimum order required, but extra cancellation can force you further.

Exam method

Predict the denominator's leading power.
Expand every numerator term to at least that order.
Combine coefficients before dividing.
Factor the common power and then take the limit.

Finding the first surviving order: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Finding the first surviving orderSelf-review

Worked example 1

\lim_{x\to0}\frac{\sin(x^2)-x^2}{x^6}.

3. Writing a scored Taylor solution

A scored Taylor solution shows the selected standard series, the substitution, the cancellation, and the final O-term. Jumping from the problem to the answer loses method marks.

O-notation records the size of omitted terms. After division by xᵏ, O(xᵐ) becomes O(xᵐ⁻ᵏ).

Exam method

State the standard expansion used.
Display substituted expansions for every component.
Combine terms on one line and identify the first survivor.
Divide, simplify the O-term, and then state the limit or polynomial.

Writing a scored Taylor solution: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Writing a scored Taylor solutionSelf-review

Worked example 1

\lim_{x\to0}\frac{e^x-1-x-x^2/2}{x^3}.

Formulas used in this topic

$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)$
$\sin x=x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^7)$
$\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6)$
$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+O(x^5)$

2WBB0 · Topic 06

Taylor expansions and hard limits

Use the supplied standard series to expose cancellation and prove local approximations.

What you need to be able to do

Recall which standard Taylor series are supplied on the exam sheet.
Expand composite expressions to the first nonzero surviving order.
Control O-terms and divide only after factoring the dominant power.

1. Reading and substituting standard series

The exam sheet supplies several standard expansions around zero. The skill is not memorizing coefficients; it is substituting a new inner expression and tracking the resulting powers.

If x is replaced by x², every power doubles. Write the new O-term explicitly so you know which orders are safe.

Exam method

Choose the supplied series matching the outer function.
Substitute the complete inner expression for x.
Expand only as far as the question's denominator or cancellation requires.
Rewrite O((xᵐ)ⁿ) as O(xᵐⁿ).

Reading and substituting standard series: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Reading and substituting standard seriesSelf-review

Worked example 1

\sin(x^2)

2. Finding the first surviving order

In a Taylor limit, the numerator is designed to cancel. Expand every term far enough to find the first nonzero coefficient, not merely to the first familiar term.

The denominator tells you the minimum order required, but extra cancellation can force you further.

Exam method

Predict the denominator's leading power.
Expand every numerator term to at least that order.
Combine coefficients before dividing.
Factor the common power and then take the limit.

Finding the first surviving order: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Finding the first surviving orderSelf-review

Worked example 1

\lim_{x\to0}\frac{\sin(x^2)-x^2}{x^6}.

3. Writing a scored Taylor solution

A scored Taylor solution shows the selected standard series, the substitution, the cancellation, and the final O-term. Jumping from the problem to the answer loses method marks.

O-notation records the size of omitted terms. After division by xᵏ, O(xᵐ) becomes O(xᵐ⁻ᵏ).

Exam method

State the standard expansion used.
Display substituted expansions for every component.
Combine terms on one line and identify the first survivor.
Divide, simplify the O-term, and then state the limit or polynomial.

Writing a scored Taylor solution: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Writing a scored Taylor solutionSelf-review

Worked example 1

\lim_{x\to0}\frac{e^x-1-x-x^2/2}{x^3}.

Formulas used in this topic

$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)$
$\sin x=x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^7)$
$\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6)$
$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+O(x^5)$

2WBB0 · Topic 06

Taylor expansions and hard limits

Use the supplied standard series to expose cancellation and prove local approximations.

What you need to be able to do

Recall which standard Taylor series are supplied on the exam sheet.
Expand composite expressions to the first nonzero surviving order.
Control O-terms and divide only after factoring the dominant power.

1. Reading and substituting standard series

The exam sheet supplies several standard expansions around zero. The skill is not memorizing coefficients; it is substituting a new inner expression and tracking the resulting powers.

If x is replaced by x², every power doubles. Write the new O-term explicitly so you know which orders are safe.

Exam method

Choose the supplied series matching the outer function.
Substitute the complete inner expression for x.
Expand only as far as the question's denominator or cancellation requires.
Rewrite O((xᵐ)ⁿ) as O(xᵐⁿ).

Reading and substituting standard series: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Reading and substituting standard seriesSelf-review

Worked example 1

\sin(x^2)

2. Finding the first surviving order

In a Taylor limit, the numerator is designed to cancel. Expand every term far enough to find the first nonzero coefficient, not merely to the first familiar term.

The denominator tells you the minimum order required, but extra cancellation can force you further.

Exam method

Predict the denominator's leading power.
Expand every numerator term to at least that order.
Combine coefficients before dividing.
Factor the common power and then take the limit.

Finding the first surviving order: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Finding the first surviving orderSelf-review

Worked example 1

\lim_{x\to0}\frac{\sin(x^2)-x^2}{x^6}.

3. Writing a scored Taylor solution

A scored Taylor solution shows the selected standard series, the substitution, the cancellation, and the final O-term. Jumping from the problem to the answer loses method marks.

O-notation records the size of omitted terms. After division by xᵏ, O(xᵐ) becomes O(xᵐ⁻ᵏ).

Exam method

State the standard expansion used.
Display substituted expansions for every component.
Combine terms on one line and identify the first survivor.
Divide, simplify the O-term, and then state the limit or polynomial.

Writing a scored Taylor solution: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Writing a scored Taylor solutionSelf-review

Worked example 1

\lim_{x\to0}\frac{e^x-1-x-x^2/2}{x^3}.

Formulas used in this topic

$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)$
$\sin x=x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^7)$
$\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6)$
$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+O(x^5)$