Fundamental Limit Laws to Know for Calculus I

Understanding Fundamental Limit Laws is key in Calculus I. These laws help simplify and evaluate limits, which are essential for analyzing functions and their behaviors. Mastering these concepts lays the groundwork for more complex calculus topics.

1. Limit of a constant

   • The limit of a constant function as x approaches any value is simply the constant itself.
   • Mathematically, lim (x → c) k = k, where k is a constant.
   • This law illustrates that constants do not change regardless of the input value.

2. Limit of x

   • The limit of the identity function as x approaches a value is that value itself.
   • Mathematically, lim (x → c) x = c.
   • This law emphasizes that the function behaves predictably as it approaches any point.

3. Sum/difference rule

   • The limit of a sum (or difference) of functions is the sum (or difference) of their limits.
   • Mathematically, lim (x → c) [f(x) ± g(x)] = lim (x → c) f(x) ± lim (x → c) g(x).
   • This rule allows for the simplification of complex limits by breaking them into simpler parts.

4. Product rule

   • The limit of a product of functions is the product of their limits.
   • Mathematically, lim (x → c) [f(x) * g(x)] = lim (x → c) f(x) * lim (x → c) g(x).
   • This rule is useful for evaluating limits involving multiplication of functions.

5. Quotient rule

   • The limit of a quotient of functions is the quotient of their limits, provided the limit of the denominator is not zero.
   • Mathematically, lim (x → c) [f(x) / g(x)] = lim (x → c) f(x) / lim (x → c) g(x), g(c) ≠ 0.
   • This rule helps in evaluating limits involving division of functions.

6. Power rule

   • The limit of a function raised to a power is the limit of the function raised to that power.
   • Mathematically, lim (x → c) [f(x)]^n = [lim (x → c) f(x)]^n, where n is a positive integer.
   • This rule simplifies the evaluation of limits involving exponentiation.

7. Root rule

   • The limit of the nth root of a function is the nth root of the limit of that function, provided the limit is non-negative.
   • Mathematically, lim (x → c) [√(f(x))] = √(lim (x → c) f(x)), f(c) ≥ 0.
   • This rule is particularly useful for limits involving square roots and higher roots.

8. Constant multiple rule

   • The limit of a constant multiplied by a function is the constant multiplied by the limit of the function.
   • Mathematically, lim (x → c) [k * f(x)] = k * lim (x → c) f(x), where k is a constant.
   • This rule allows for the extraction of constants from limits, simplifying calculations.

9. Limits of composite functions

   • The limit of a composite function can be evaluated using the limit of the inner function.
   • Mathematically, if g is continuous at c, then lim (x → c) f(g(x)) = f(lim (x → c) g(x)).
   • This rule is essential for dealing with nested functions in limit problems.

10. Squeeze theorem

    • The Squeeze theorem states that if a function is "squeezed" between two other functions that have the same limit at a point, then it also has that limit.
    • Mathematically, if h(x) ≤ f(x) ≤ g(x) and lim (x → c) h(x) = lim (x → c) g(x) = L, then lim (x → c) f(x) = L.
    • This theorem is particularly useful for finding limits of functions that are difficult to evaluate directly.