Limit of a Constant:
\( \lim \limits_{x \to a} c = c \)
\( \lim \limits_{x \to a} x = a \)
\(\lim \limits_{x \to a} \frac{x^n-a^n}{x-a} =na^{n-1} \)
\(\lim \limits_{x \to a} \frac{x^m-a^m}{x^n-a^n} =\frac{m^{m-n}}{n} \)
\(\lim \limits_{x \to 0} \frac{a^x-1}{x} = \ln a \)
Exponential Functions:
\( \lim \limits_{x \to a} e^x = e^a \)
\( \lim \limits_{x \to 0} \frac{e^x-1}{x} = 1 \)
Logarithmic Functions:
\( \lim \limits_{x \to a} \ln(x) = \ln(a) \)
\( \lim \limits_{x \to 0} \frac{\ln(1+x)}{x} = 1 \)
Trigonometric Functions:
\( \lim \limits_{x \to c} \sin(x) = \sin(c) \)
\( \lim \limits_{x \to c} \cos(x) = \cos(c) \)
\( \lim \limits_{x \to c} \sec(x) = \sec(c) \)
\( \lim \limits_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1 \)
\( \lim \limits_{\theta \to 0} \sin(\theta) = 0 \)
\( \lim \limits_{\theta \to 0} \cos(\theta) = 1 \)
\( \lim \limits_{x \to 0} \frac{1 – \cos(x)}{x} = 0 \)
Infinity Rules:
\( \lim \limits_{x \to \infty} \frac{1}{x} = 0 \)
\( \lim \limits_{x \to \infty} \frac{1}{x^2} = 0 \)
\( \lim \limits_{x \to \infty} e^{-x} = 0 \)
\( \lim \limits_{x \to \infty} \ln(x) = \infty \)
Special Limits:
\( \lim \limits_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e \)
\( \lim \limits_{x \to 0^+} x^x = 1 \)
Sum/Diff Rule:
\( \lim \limits_{x \to a} \left[ f(x) \pm g(x) \right] = \lim \limits_{x \to a} f(x) \pm \lim \limits_{x \to a} g(x)\)
Product Rule:
\( \lim \limits_{x \to a} \left[ f(x) \cdot g(x) \right] = \left( \lim \limits_{x \to a} f(x) \right) \cdot \left( \lim \limits_{x \to a} g(x) \right) \)
Quotient Rule:
\( \lim \limits_{x \to a} \left[ \frac{f(x)}{g(x)} \right] = \frac{\lim \limits_{x \to a} f(x)}{\lim \limits_{x \to a} g(x)} \)
Power Rule:
\( \lim \limits_{x \to a} \left[ f(x) \right]^n = \left( \lim \limits_{x \to a} f(x) \right)^n \)
Root Rule:
\( \lim \limits_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim \limits_{x \to a} f(x)} \quad \text{if } f(x) \geq 0 \)
L’Hôpital’s Rule:
If \( \lim \limits_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\pm \infty}{\pm \infty} \), then
\( \lim \limits_{x \to a} \frac{f(x)}{g(x)} = \lim \limits_{x \to a} \frac{f'(x)}{g'(x)} \),
provided the derivatives exist and the limit on the right exists.
Continuity and Limits:
If ( f(x) ) is continuous at ( x = a ), then \( \lim \limits_{x \to a} f(x) = f(a) \).