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⬅ Back to Differential Calculus Topics

Limits and Continuity

Limits describe the behavior of a function as the input approaches a value. Continuity means the limit equals the function value at that point.

$$\lim_{x \to a} f(x) = L$$
$$\text{Continuous at } x=a \iff \begin{cases} f(a) \text{ is defined} \\ \lim_{x \to a} f(x) \text{ exists} \\ \lim_{x \to a} f(x) = f(a) \end{cases} $$

Types of discontinuities: removable, jump, infinite.

Definition of the Derivative

The derivative measures instantaneous rate of change and slope of the tangent line.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
$$f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$$

Tangent and Normal Lines

1. Equation of the tangent line at $x=a$:

$$y - f(a) = f'(a)(x - a)$$

2. Equation of the normal line at $x=a$:

$$y - f(a) = -\frac{1}{f'(a)}(x - a)$$

Linear Approximation and Differentials — Key Formulas

$$L(x) = f(a) + f'(a)(x-a)$$
$$dy = f'(x)\, dx$$
$$\Delta y \approx dy$$

Evaluating a Limit by Factoring — CE Board

Evaluate: $\displaystyle\lim_{x\to 3}\frac{x^2 - 9}{x - 3}$

Direct substitution gives $0/0$. Factor the numerator:

$\lim_{x\to 3}\dfrac{(x-3)(x+3)}{x-3} = \lim_{x\to 3}(x+3) = 3 + 3 = \boxed{6}$

Derivative from Definition — CE Board

Using the limit definition $f'(x) = \displaystyle\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$, find $f'(x)$ if $f(x) = x^2$.

$f'(x) = \lim_{h\to 0}\dfrac{(x+h)^2 - x^2}{h} = \lim_{h\to 0}\dfrac{2xh + h^2}{h} = \lim_{h\to 0}(2x + h)$
$\boxed{f'(x) = 2x}$

Equation of Tangent Line — CE Board

Find the equation of the tangent line to $y = x^2$ at the point $(2,\,4)$.

Slope: $y' = 2x$, so at $x = 2$: $m = 4$.

$y - 4 = 4(x - 2)$
$\boxed{y = 4x - 4}$

Equation of Normal Line — CE Board

Find the equation of the normal line to $y = x^3$ at the point $(1,\,1)$.

Slope of tangent: $y' = 3x^2 = 3$ at $x = 1$.

Slope of normal (negative reciprocal): $m_n = -\dfrac{1}{3}$.

$y - 1 = -\dfrac{1}{3}(x - 1)$
$3y - 3 = -(x - 1) \implies \boxed{x + 3y = 4}$

Fundamental Trigonometric Limit — CE Board

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

This is the standard trigonometric limit, proved via the squeeze theorem or L'Hôpital's Rule:

$\lim_{x\to 0}\dfrac{\sin x}{x} = \boxed{1}$

This result is used extensively in derivative proofs for $\sin x$.

Corollary: $\displaystyle\lim_{x\to 0}\dfrac{\tan x}{x} = 1$ and $\displaystyle\lim_{x\to 0}\dfrac{1-\cos x}{x} = 0$

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