Limits and Derivatives

The One Thing

Limits describe the value approached by a function near an input, while derivatives use limits to measure instantaneous change. The derivative at a point is both the instantaneous rate of change and the slope of the tangent to the graph.

Definitions and Results

• Limit: The number a function approaches as the independent variable approaches a specified value, whether or not the function is defined at that value. The notation is ${\lim }_{x\to a}f(x)$.
• Left-hand limit: The value approached when $x$ approaches $a$ from values smaller than $a$, written ${\lim }_{x\to a^{-}}f(x)$.
• Right-hand limit: The value approached when $x$ approaches $a$ from values greater than $a$, written ${\lim }_{x\to a^{+}}f(x)$.
• Existence of a limit: ${\lim }_{x\to a}f(x)$ exists only when the left-hand and right-hand limits are equal and finite.
• Continuity at a point: A function is continuous at $x=a$ if $f(a)$ exists, ${\lim }_{x\to a}f(x)$ exists, and ${\lim }_{x\to a}f(x)=f(a)$.
• Indeterminate form: An expression such as $0/0$ does not determine a limit directly and requires further simplification.
• Standard trigonometric limits: When angles are measured in radians,

$$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1,\underset{x\to 0}{\lim }\frac{\tan x}{x}=1.$$ Also, $$\underset{x\to 0}{\lim }\frac{1-\cos x}{x}=0.$$

• Scaled trigonometric limits: For a constant $a$, with angles measured in radians,

$$\underset{x\to 0}{\lim }\frac{\sin (ax)}{x}=a,\underset{x\to 0}{\lim }\frac{\tan (ax)}{x}=a.$$ Useful equivalent forms are $$\underset{x\to 0}{\lim }\frac{\sin (ax)}{\sin (bx)}=\frac{a}{b},\underset{x\to 0}{\lim }\frac{\tan (ax)}{\tan (bx)}=\frac{a}{b},$$ when the expressions are defined near zero.

• Derivative: The instantaneous rate of change of a function with respect to its variable.
• Derivative from first principles: The derivative of $f(x)$ at $x=a$ is

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h},$$ provided the limit exists.

• Geometrical meaning of derivative: $f^{\prime }(a)$ is the slope of the tangent to the graph at $x=a$.
• Differentiability: A function is differentiable at a point if its derivative exists there. Differentiability implies continuity, although continuity does not necessarily imply differentiability.
• Derivative as a function: $f^{\prime }(x)$ gives the derivative at every point where the derivative exists.
• Basic derivative rules:

$$\frac{d}{dx}(c)=0,\frac{d}{dx}(x^n)=n{x}^{n-1},$$ for suitable real or integer values of $n$ in the introductory setting; $$\frac{d}{dx}[u\pm v]=u^{\prime }\pm v^{\prime },\frac{d}{dx}[cu]=c{u}^{\prime }.$$

• Trigonometric derivatives: When angles are measured in radians,

$$\frac{d}{dx}(\sin x)=\cos x,\frac{d}{dx}(\cos x)=-\sin x,$$ and $$\frac{d}{dx}(\tan x)={\sec }^{2}x$$ wherever $\tan x$ is defined.

• Tangent equation: The tangent to $y=f(x)$ at $(a,f(a))$ is

$$y-f(a)=f^{\prime }(a)(x-a).$$

• Normal equation: When $f^{\prime }(a)\ne 0$, the normal at $(a,f(a))$ is

$$y-f(a)=-\frac{1}{f^{\prime }(a)}(x-a).$$

• Physical interpretation: If position is a function of time, its derivative represents instantaneous velocity.
• Continuity and differentiability: A function may be continuous but not differentiable at a point, for example at a sharp corner.

Worked Methods

Evaluating a limit by direct substitution

• Identify the limiting value $a$.
• Substitute $x=a$ into $f(x)$.
• For a polynomial function, direct substitution is valid because

$$\underset{x\to a}{\lim }f(x)=f(a).$$

Evaluating a limit that gives $0/0$

• Substitute the limiting value.
• If the result is $0/0$, recognise it as an indeterminate form.
• Simplify the expression using factorisation, rationalisation, or a standard limit result.
• Evaluate the simplified expression.

The original $0/0$ expression does not itself determine the limit; algebraic simplification is required.

Using standard trigonometric limits

• Rewrite the expression so that it contains

$$\frac{\sin x}{x}=1\text{or}\frac{\tan x}{x}=1.$$

• Ensure that angles are measured in radians.
• Apply the relevant standard result.

For scaled expressions, use $$\underset{x\to 0}{\lim }\frac{\sin (ax)}{x}=a\text{and}\underset{x\to 0}{\lim }\frac{\tan (ax)}{x}=a.$$

Establishing whether a two-sided limit exists

• Calculate the left-hand limit ${\lim }_{x\to a^{-}}f(x)$.
• Calculate the right-hand limit ${\lim }_{x\to a^{+}}f(x)$.
• Compare the results.
• The two-sided limit exists exactly when both one-sided limits exist and are equal and finite.

Finding a derivative from first principles

• Start with

$$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$$

• Substitute the function $f$.
• Expand and simplify the numerator.
• Cancel any common factor of $h$, where appropriate.
• Evaluate the limit as $h\to 0$.

This definition expresses the derivative as the limiting value of the average rate of change over increasingly small intervals.

Finding derivatives using basic rules

• Identify constants, powers, sums, differences, and constant multiples.
• Apply

$$\frac{d}{dx}(c)=0\text{and}\frac{d}{dx}(x^n)=n{x}^{n-1}.$$

• Differentiate sums and differences term by term:

$$\frac{d}{dx}[u\pm v]=u^{\prime }\pm v^{\prime }.$$

• Preserve constant multiples:

$$\frac{d}{dx}[cu]=c{u}^{\prime }.$$

• For trigonometric terms, use

$$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x.$$

Finding a tangent and normal

• Find the point $(a,f(a))$.
• Calculate the derivative $f^{\prime }(x)$.
• Evaluate $f^{\prime }(a)$ to obtain the tangent slope.
• Substitute into

$$y-f(a)=f^{\prime }(a)(x-a).$$

• If $f^{\prime }(a)\ne 0$, use the reciprocal negative slope for the normal:

$$y-f(a)=-\frac{1}{f^{\prime }(a)}(x-a).$$

Where It Goes Wrong

• Treating $f(a)$ as the limit without checking nearby function behaviour; a limit concerns values approached near $a$, whether or not the function is defined at $a$.
• Assuming a two-sided limit exists without comparing ${\lim }_{x\to a^{-}}f(x)$ and ${\lim }_{x\to a^{+}}f(x)$.
• Treating $0/0$ as an answer rather than an indeterminate form requiring factorisation, rationalisation, or a standard limit result.
• Applying the standard trigonometric limits when angles are not measured in radians.
• Confusing the derivative with an average rate of change; the derivative requires the limiting process $h\to 0$.
• Assuming continuity guarantees differentiability; a continuous function can have a sharp corner and therefore fail to be differentiable there.

What Gets Asked

• Evaluate a limit by direct substitution for a polynomial.
• Simplify and evaluate a limit that initially produces $0/0$.
• Use standard trigonometric limits, including scaled forms and equivalent sine or tangent ratios.
• Determine whether a two-sided limit exists from left-hand and right-hand limits.
• Test continuity at $x=a$.
• Find $f^{\prime }(a)$ from first principles.
• Differentiate constants, powers, sums, differences, constant multiples, $\sin x$, $\cos x$, and $\tan x$.
• Interpret a derivative as an instantaneous rate of change, including instantaneous velocity when position depends on time.
• Find the slope and equation of the tangent to $y=f(x)$ at $x=a$.
• Find the equation of the normal when $f^{\prime }(a)\ne 0$.
• Explain the relationship between continuity and differentiability.