Calculus Limits and Derivatives Review

Flashcards covering core calculus concepts including limit conditions, continuity, theorems, and differentiation rules.

Condition for Limit Existence

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$

Conditions for Continuity at $x = a$

1. $f(a)$ is defined; 2. \lim_{x \to a} f(x) exists; 3. \lim_{x \to a} f(x) = f(a)

Squeeze Theorem

If $f(x) \le g(x) \le h(x)$ and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ & $f(b)$, then $f(c) = N$ for some $c \in (a, b)$.

Limit Definition of $f'(a)$

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

Tangent Line Equation at $x = a$

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

Power Rule

$$(x^n)' = n x^{n-1}$$

Product Rule

$$[f \cdot g]' = f' g + f g'$$

Quotient Rule

$$[f / g]' = \frac{f' g - f g'}{g^2}$$

Differentiability and Continuity Relationship

Differentiability implies continuity ($f$ diff at $a \implies f$ cont at $a$).