Calculus
What is Calculus?
The value that a function or sequence 'approaches' as the input or index approaches some value.
Key formula / rule: Limit Definition
Key points
- Understand the definition and calculation of limits.
- Determine the continuity of functions at a point and over an interval.
- Assess the differentiability of functions.
- Apply limit, continuity, and differentiability concepts to analyze function behavior.
Common exam trap
Confusing limits with function values at a point.
Definitions
- Term
Limit
- Meaning
The value that a function or sequence 'approaches' as the input or index approaches some value.
- Term
Continuity
- Meaning
A function is continuous at a point if its graph can be drawn through the point without lifting the pen; formally, if the limit exists, the function is defined at the point, and the limit equals the function's value.
- Term
Differentiability
- Meaning
A function is differentiable at a point if its derivative exists at that point, meaning the function has a unique, non-vertical tangent line.
- Term
Indeterminate Form
- Meaning
An expression (like 0/0 or ∞/∞) that arises in the evaluation of limits, requiring further analysis (e.g., L'Hôpital's Rule) to determine the limit's value.
Learning objectives
Understand the definition and calculation of limits.
Determine the continuity of functions at a point and over an interval.
Assess the differentiability of functions.
Apply limit, continuity, and differentiability concepts to analyze function behavior.
Formulae
- Name
Limit Definition
- Note
f(x) approaches L as x approaches c.
- Expression
limx→c f(x) = L
- Name
Continuity Condition
- Note
Requires f(c) to be defined, limit to exist, and limit to equal f(c).
- Expression
limx→c f(x) = f(c)
- Name
L'Hôpital's Rule
- Note
Applicable for indeterminate forms 0/0 or ∞/∞.
- Expression
limx→c \frac{f(x)}{g(x)} = limx→c \frac{f'(x)}{g'(x)}
Prerequisites
Basic algebra (functions, equations, inequalities).
Understanding of function graphs.
Knowledge of basic trigonometric and exponential functions.
Common mistakes
Confusing limits with function values at a point.
Assuming continuity implies differentiability.
Errors in applying L'Hôpital's Rule (e.g., when the form is not indeterminate).
Incorrectly evaluating limits involving ∞ or piecewise functions.
Keywords
Limit
Continuity
Differentiability
L'Hôpital's Rule
Epsilon-Δ
Indeterminate Form
Rate of Change
Practice preview
Find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1 with respect to x.…
easy
Find the local maximum value of the function f(x) = x^3 - 6x^2 + 9x + 1.…
medium
Evaluate the definite integral: integral(from 0 to pi/2) (sin^2(x)) dx.…
medium