2WBB0Calculus I

2WBB0 · Topic 04

Limits, continuity, and differentiability

Choose the right limit method and handle piecewise functions without guessing.

What you need to be able to do

Classify a limit before choosing algebra, squeeze, l'Hôpital, or Taylor.
Use dominant terms and bounded oscillation at infinity.
Test continuity and differentiability of piecewise functions from both sides.

1. The limit decision tree

Substitution is always the first test. If it gives a finite value, the problem is finished. If it gives an indeterminate form, identify the form before selecting a technique.

Factor and rationalize for algebraic cancellations; use standard small-angle limits or Taylor near zero; reserve l'Hôpital for verified quotient forms.

Exam method

Substitute and write the resulting form.
Simplify algebraically before differentiating anything.
Use the lowest-cost valid method: factor, rationalize, squeeze, standard limit, Taylor, then l'Hôpital.
After transforming, substitute again and state the final value.

The limit decision tree: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

The limit decision treeSelf-review

Worked example 1

\lim_{x\to0}\frac{\sqrt{x+2}-\sqrt2}{x}.

2. Dominant terms and bounded oscillation

At infinity, polynomial growth dominates bounded sine and cosine terms. Divide by the largest power or growth factor present.

The squeeze theorem is ideal when a bounded factor is multiplied by something tending to zero.

Exam method

Identify the dominant growth in numerator and denominator.
Divide every term by that growth factor.
Use boundedness to eliminate sine or cosine divided by an unbounded quantity.
Do not replace an oscillating term by a limit it does not have.

Dominant terms and bounded oscillation: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

Dominant terms and bounded oscillationSelf-review

Worked example 1

\lim_{x\to\infty}\frac{x-2\sin x}{x+3\cos x}.

3. Continuity versus differentiability

Continuity at a join requires the left limit, right limit, and defined value to agree. Differentiability additionally requires equal one-sided derivatives.

A function can be continuous with a corner and therefore fail to be differentiable. A differentiable function cannot be discontinuous.

Exam method

Compute the left value or limit at the join.
Compute the right value or limit and solve continuity parameters first.
Differentiate each branch and compare one-sided derivatives.
State continuity and differentiability as separate conclusions.

Continuity versus differentiability: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

Continuity versus differentiabilitySelf-review

Worked example 1

c

Formulas used in this topic

$\lim_{x\to a}\frac{f(x)}{g(x)}\ \text{uses l'Hôpital only for }0/0\text{ or }\infty/\infty$
$|\sin x|\le1,\qquad |\cos x|\le1$
$f\text{ continuous at }a\iff\lim_{x\to a}f(x)=f(a)$
$f\text{ differentiable at }a\Longrightarrow f\text{ continuous at }a$

2WBB0 · Topic 04

Limits, continuity, and differentiability

Choose the right limit method and handle piecewise functions without guessing.

What you need to be able to do

Classify a limit before choosing algebra, squeeze, l'Hôpital, or Taylor.
Use dominant terms and bounded oscillation at infinity.
Test continuity and differentiability of piecewise functions from both sides.

1. The limit decision tree

Substitution is always the first test. If it gives a finite value, the problem is finished. If it gives an indeterminate form, identify the form before selecting a technique.

Factor and rationalize for algebraic cancellations; use standard small-angle limits or Taylor near zero; reserve l'Hôpital for verified quotient forms.

Exam method

Substitute and write the resulting form.
Simplify algebraically before differentiating anything.
Use the lowest-cost valid method: factor, rationalize, squeeze, standard limit, Taylor, then l'Hôpital.
After transforming, substitute again and state the final value.

The limit decision tree: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

The limit decision treeSelf-review

Worked example 1

\lim_{x\to0}\frac{\sqrt{x+2}-\sqrt2}{x}.

2. Dominant terms and bounded oscillation

At infinity, polynomial growth dominates bounded sine and cosine terms. Divide by the largest power or growth factor present.

The squeeze theorem is ideal when a bounded factor is multiplied by something tending to zero.

Exam method

Identify the dominant growth in numerator and denominator.
Divide every term by that growth factor.
Use boundedness to eliminate sine or cosine divided by an unbounded quantity.
Do not replace an oscillating term by a limit it does not have.

Dominant terms and bounded oscillation: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

Dominant terms and bounded oscillationSelf-review

Worked example 1

\lim_{x\to\infty}\frac{x-2\sin x}{x+3\cos x}.

3. Continuity versus differentiability

Continuity at a join requires the left limit, right limit, and defined value to agree. Differentiability additionally requires equal one-sided derivatives.

A function can be continuous with a corner and therefore fail to be differentiable. A differentiable function cannot be discontinuous.

Exam method

Compute the left value or limit at the join.
Compute the right value or limit and solve continuity parameters first.
Differentiate each branch and compare one-sided derivatives.
State continuity and differentiability as separate conclusions.

Continuity versus differentiability: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

Continuity versus differentiabilitySelf-review

Worked example 1

c

Formulas used in this topic

$\lim_{x\to a}\frac{f(x)}{g(x)}\ \text{uses l'Hôpital only for }0/0\text{ or }\infty/\infty$
$|\sin x|\le1,\qquad |\cos x|\le1$
$f\text{ continuous at }a\iff\lim_{x\to a}f(x)=f(a)$
$f\text{ differentiable at }a\Longrightarrow f\text{ continuous at }a$

2WBB0 · Topic 04

Limits, continuity, and differentiability

Choose the right limit method and handle piecewise functions without guessing.

What you need to be able to do

Classify a limit before choosing algebra, squeeze, l'Hôpital, or Taylor.
Use dominant terms and bounded oscillation at infinity.
Test continuity and differentiability of piecewise functions from both sides.

1. The limit decision tree

Substitution is always the first test. If it gives a finite value, the problem is finished. If it gives an indeterminate form, identify the form before selecting a technique.

Factor and rationalize for algebraic cancellations; use standard small-angle limits or Taylor near zero; reserve l'Hôpital for verified quotient forms.

Exam method

Substitute and write the resulting form.
Simplify algebraically before differentiating anything.
Use the lowest-cost valid method: factor, rationalize, squeeze, standard limit, Taylor, then l'Hôpital.
After transforming, substitute again and state the final value.

The limit decision tree: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

The limit decision treeSelf-review

Worked example 1

\lim_{x\to0}\frac{\sqrt{x+2}-\sqrt2}{x}.

2. Dominant terms and bounded oscillation

At infinity, polynomial growth dominates bounded sine and cosine terms. Divide by the largest power or growth factor present.

The squeeze theorem is ideal when a bounded factor is multiplied by something tending to zero.

Exam method

Identify the dominant growth in numerator and denominator.
Divide every term by that growth factor.
Use boundedness to eliminate sine or cosine divided by an unbounded quantity.
Do not replace an oscillating term by a limit it does not have.

Dominant terms and bounded oscillation: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

Dominant terms and bounded oscillationSelf-review

Worked example 1

\lim_{x\to\infty}\frac{x-2\sin x}{x+3\cos x}.

3. Continuity versus differentiability

Continuity at a join requires the left limit, right limit, and defined value to agree. Differentiability additionally requires equal one-sided derivatives.

A function can be continuous with a corner and therefore fail to be differentiable. A differentiable function cannot be discontinuous.

Exam method

Compute the left value or limit at the join.
Compute the right value or limit and solve continuity parameters first.
Differentiate each branch and compare one-sided derivatives.
State continuity and differentiability as separate conclusions.

Continuity versus differentiability: worked examples

2 questions

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

Question 1/2 · 0 reviews recorded

Continuity versus differentiabilitySelf-review

Worked example 1

c

Formulas used in this topic

$\lim_{x\to a}\frac{f(x)}{g(x)}\ \text{uses l'Hôpital only for }0/0\text{ or }\infty/\infty$
$|\sin x|\le1,\qquad |\cos x|\le1$
$f\text{ continuous at }a\iff\lim_{x\to a}f(x)=f(a)$
$f\text{ differentiable at }a\Longrightarrow f\text{ continuous at }a$