Properties of Limits

Basic Properties

• Constant Law: The limit of a constant function is the constant value itself.

  $$\lim_{{x \to a}} c = c$$

• Identity Law: The limit of the identity function as $x$ approaches $a$ is $a$.

  $$\lim_{{x \to a}} x = a$$

Operations with Limits

Sum and Difference Law

• The limit of the sum (or difference) of two functions is the sum (or difference) of their respective limits.

  $$\lim_{{x \to a}} [f(x) \pm g(x)] = \lim_{{x \to a}} f(x) \pm \lim_{{x \to a}} g(x)$$

Constant Multiple Law

• The limit of a constant multiplied by a function is the constant multiplied by the limit of the function.

  $$\lim_{{x \to a}} [c \cdot f(x)] = c \cdot \lim_{{x \to a}} f(x)$$

Product Law

• The limit of a product of two functions is the product of their limits.

  $$\lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)$$

Quotient Law

• The limit of the quotient of two functions is the quotient of their limits, given that the limit of the denominator is not zero.

  $$\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)}$$

Note: It is essential that $\lim_{{x \to a}} g(x) \neq 0$.

Power Law

• The limit of a function raised to a power is the limit of the function raised to that power.

  $$\lim_{{x \to a}} [f(x)]^n = \left(\lim_{{x \to a}} f(x)\right)^n$$

Root Law

• The limit of a root of a function is the root of the limit of the function.

  $$\lim_{{x \to a}} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{{x \to a}} f(x)}$$

Use in Continuity

• These properties aid in determining the continuity of functions at given points, helping to identify if the function behaves as expected without interruption or jumps.

Additional Considerations

• It is important to note the existence of limits. For any of the laws to apply, the limits of the individual functions involved must exist.
• In the case of infinite limits or limits at infinity, these laws are applied with additional caution, especially considering the behavior of functions as they grow without bound.

References

• calculus_cheat_sheet_limits | tutorial.math.lamar.edu