Limit Rules in Calculus with Examples
Basic Limit Rules
1. Constant Rule
Rule: \( \lim_{{x \to c}} k = k \)
Example:
\( \lim_{{x \to 4}} 7 = 7 \)
2. Identity Rule
Rule: \( \lim_{{x \to c}} x = c \)
Example:
\( \lim_{{x \to 5}} x = 5 \)
3. Constant Multiple Rule
Rule: \( \lim_{{x \to c}} [k \cdot f(x)] = k \cdot \lim_{{x \to c}} f(x) \)
Example:
\( \lim_{{x \to 3}} 4x = 4 \cdot \lim_{{x \to 3}} x = 4 \cdot 3 = 12 \)
4. Sum Rule
Rule: \( \lim_{{x \to c}} [f(x) + g(x)] = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x) \)
Example:
\( \lim_{{x \to 2}} [x^2 + 3x] = \lim_{{x \to 2}} x^2 + \lim_{{x \to 2}} 3x = 2^2 + 3(2) = 10 \)
5. Difference Rule
Rule: \( \lim_{{x \to c}} [f(x) - g(x)] = \lim_{{x \to c}} f(x) - \lim_{{x \to c}} g(x) \)
Example:
\( \lim_{{x \to 1}} [x^3 - 2x] = \lim_{{x \to 1}} x^3 - \lim_{{x \to 1}} 2x = 1^3 - 2(1) = -1 \)
Power and Root Rules
8. Power Rule
Rule: \( \lim_{{x \to c}} [f(x)]^n = \left[\lim_{{x \to c}} f(x)\right]^n \)
Example:
\( \lim_{{x \to 2}} (x^3) = \left(\lim_{{x \to 2}} x\right)^3 = 2^3 = 8 \)
9. Root Rule
Rule: \( \lim_{{x \to c}} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{{x \to c}} f(x)} \)
Example:
\( \lim_{{x \to 9}} \sqrt{x} = \sqrt{\lim_{{x \to 9}} x} = \sqrt{9} = 3 \)
Special Cases
13. Squeeze Theorem
Rule: If \( h(x) \leq f(x) \leq g(x) \) and \( \lim_{{x \to c}} h(x) = \lim_{{x \to c}} g(x) = L \), then \( \lim_{{x \to c}} f(x) = L \).
Example:
\( \lim_{{x \to 0}} x^2 \sin\left(\frac{1}{x}\right) = 0 \)
14. L’Hôpital’s Rule
Rule: \( \lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} \), if the original limit is indeterminate.
Example:
\( \lim_{{x \to 0}} \frac{\sin(x)}{x} = \lim_{{x \to 0}} \frac{\cos(x)}{1} = 1 \)
Trigonometric Limits
15. Basic Trigonometric Limit
Rule: \( \lim_{{x \to 0}} \frac{\sin(x)}{x} = 1 \)
Example:
\( \lim_{{x \to 0}} \frac{\sin(2x)}{2x} = 1 \)