How to Find Limits in Calculus
Evaluating limits is the starting point of calculus. When plugging in a number gives you 0/0, these four methods will help you solve almost any limit problem.
1. Direct Substitution (Try This First)
The simplest technique is to plug the target number directly into x. If the function is continuous at that point, the result is your answer.
2. Factoring and Canceling (For 0/0 Forms)
If direct substitution yields 0/0, you have an indeterminate form. There is usually a removable "hole" in the graph caused by a common factor.
Example: Evaluate lim (x → 2) (x² - 4) / (x - 2).
- Plugging in
x = 2gives(4 - 4)/(2 - 2) = 0/0. - Factor the numerator:
x² - 4 = (x - 2)(x + 2). - Cancel the common
(x - 2)factor:lim (x → 2) (x + 2). - Now substitute
x = 2:2 + 2 = 4.
3. Multiplying by the Conjugate (For Square Roots)
When an expression has radical terms (square roots) producing 0/0, multiply both the numerator and denominator by the conjugate (change the sign between the terms).
For example, if you have (√(x + 1) - 1) / x as x → 0, multiply by (√(x + 1) + 1). The square roots eliminate, leaving an algebraic factor that cancels cleanly.
4. L'Hôpital's Rule
For stubborn fractions producing 0/0 or ∞/∞, differentiate the top and bottom separately:
Try it Live: Visualizing a Removable Discontinuity
Try plotting f(x) = (x^2 - 4)/(x - 2) below. The line looks solid at y = x + 2, but if you look at x = 2, the true limit approaches exactly y = 4!
Frequently Asked Questions
What is a limit in calculus?
A limit is the value that a function approaches as the input (x) gets closer and closer to a particular number, regardless of whether the function is actually defined at that exact point.
What does an indeterminate form 0/0 mean?
An indeterminate form like 0/0 does not mean undefined or zero. It means direct substitution failed and further algebraic work (such as factoring, canceling, rationalizing, or L'Hôpital's Rule) is needed to reveal the true limit value.
When does a limit fail to exist (DNE)?
A two-sided limit does not exist (DNE) if: 1) the left-hand limit and right-hand limit approach different values, 2) the function oscillates infinitely, or 3) the function approaches infinity from one or both sides.
What is L'Hôpital's Rule?
L'Hôpital's Rule states that if evaluating a limit lim(x→a) [f(x)/g(x)] produces 0/0 or ±∞/±∞, you can differentiate the numerator and denominator independently: lim [f'(x)/g'(x)].