Limits Cheat Sheet

What is a Limit?

A limit describes the behavior of a function as its input approaches a particular value. It allows us to analyze functions at points where they might be undefined or to determine their long-term behavior.

The limit of f(x) as x approaches a is written as:

This means that as x values get arbitrarily close to a (but not equal to a), the corresponding values of f(x) get arbitrarily close to L.

Basic Limit Properties

If lim_xa f(x) = L and lim_xa g(x) = M, then:

• Sum Rule: lim_xa [f(x) + g(x)] = L + M
• Difference Rule: lim_xa [f(x) - g(x)] = L - M
• Product Rule: lim_xa [f(x) g(x)] = L M
• Quotient Rule: lim_xa [f(x) / g(x)] = L / M (provided M 0)
• Constant Multiple Rule: lim_xa [cf(x)] = cL for any constant c
• Power Rule: lim_xa [f(x)]ⁿ = Lⁿ for any positive integer n
• Root Rule: lim_xa ⁿ[f(x)] = ⁿL for any odd integer n (and even n when L > 0)

Direct Substitution

For many functions, particularly polynomials, rational functions, trigonometric functions, exponential functions, and logarithmic functions, you can evaluate the limit by direct substitution:

Direct substitution works when the function is continuous at point a.

Techniques for Evaluating Limits

When direct substitution results in an indeterminate form like 0/0, /, or 0, other techniques must be used:

1. Factoring

Factor and simplify the expression when possible to eliminate the indeterminate form.

2. Conjugate Multiplication

Multiply by the conjugate to eliminate square roots.

3. Special Limits

Use known special limits:

4. L'Hpital's Rule

When direct substitution yields 0/0 or /, if the functions are differentiable:

5. Squeeze Theorem

If g(x) f(x) h(x) for all x near a (except possibly at a) and lim_xa g(x) = lim_xa h(x) = L, then lim_xa f(x) = L.

One-Sided Limits

Sometimes it's important to consider the direction from which x approaches a:

• Left-hand limit: lim_xa f(x) = L means x approaches a from values less than a
• Right-hand limit: lim_xa f(x) = L means x approaches a from values greater than a

For the two-sided limit lim_xa f(x) to exist and equal L, both one-sided limits must exist and equal L.

Infinite Limits

Infinite limits indicate that a function grows without bound as x approaches a specific value:

Vertical asymptotes often occur at points where limits are infinite.

Limits at Infinity

Limits at infinity describe the behavior of a function as x grows arbitrarily large:

• Rational Functions: For lim_x p(x)/q(x) where p and q are polynomials:
  • If degree(p) < degree(q), the limit is 0
  • If degree(p) = degree(q), the limit is the ratio of leading coefficients
  • If degree(p) > degree(q), the limit is , -, or does not exist
• Exponential Functions:
  • lim_x e^kx = for k > 0
  • lim_x e^kx = 0 for k < 0
• Logarithmic Functions:
  • lim_x ln(x) =
  • lim_x0 ln(x) = -

Continuity

A function f is continuous at a number a if:

1. f(a) is defined
2. lim_xa f(x) exists
3. lim_xa f(x) = f(a)

Common continuous functions:

• Polynomials are continuous everywhere
• Rational functions are continuous on their domains
• Trig, exponential, and log functions are continuous on their domains

The Intermediate Value Theorem: If f is continuous on [a,b] and N is any number between f(a) and f(b), then there exists at least one c in (a,b) such that f(c) = N.

Common Limit Formulas

Tips for Evaluating Limits

1. Always try direct substitution first
2. If you get 0/0, try factoring or rationalizing
3. For limits involving trigonometric functions, use trigonometric identities and special limits
4. For limits involving exponentials or logarithms, consider using properties of these functions
5. If L'Hpital's Rule applies, check if differentiating the numerator and denominator simplifies the expression
6. For limits at infinity of rational functions, compare the degrees of the numerator and denominator
7. Graph the function to understand its behavior near the limit point
8. Check for one-sided limits when the two-sided limit doesn't exist