Quotient law for limits

Definition

The Quotient Law for limits states that the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. Mathematically, if $\lim_{{x \to c}} f(x) = L$ and $\lim_{{x \to c}} g(x) = M$ with $M \neq 0$, then $\lim_{{x \to c}} \frac{f(x)}{g(x)} = \frac{L}{M}$.

5 Must Know Facts For Your Next Test

1. The Quotient Law for limits requires that the limit of the denominator is not zero.
2. This law can be applied to both one-sided and two-sided limits.
3. If either the numerator or denominator does not have a limit, then you cannot apply this law directly.
4. When applying this law, ensure that both functions involved have well-defined limits at point c.
5. Common pitfalls include not checking if $M$ (limit of the denominator) is non-zero before applying the Quotient Law.

Review Questions

• What condition must be satisfied for the Quotient Law for limits to apply?
• How do you apply the Quotient Law when given $\lim_{{x \to c}} f(x)$ and $\lim_{{x \to c}} g(x)$?
• Explain why it is important to check that $\lim_{{x \to c}} g(x) \neq 0$ when using the Quotient Law.

Related terms

Limit: The value that a function approaches as the input approaches some value.

Sum Law for Limits: $\lim_{{x \to c}} [f(x) + g(x)] = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x)$

Product Law for Limits: $\lim_{{x \to c}} [f(x) \cdot g(x)] = (\lim_{{x \to c}} f(x)) \cdot (\lim_{{x \to c}} g(x))$