What is the Limit Formula?

The limit formula is expressed using the notation below and defines the value a function approaches as its input approaches a certain value.

The core concept of a limit describes the behavior of a function f(x) as x gets arbitrarily close to a specific value a. It essentially asks, "What value does f(x) get closer and closer to as x approaches a?".

Mathematically, the limit is represented as:

lim_x→a f(x) = L

Explanation of the Limit Formula:

• lim: This is the abbreviation for "limit."
• x → a: This signifies that "x approaches a." In other words, we're looking at what happens to the function as x gets closer and closer to the value a, but x never actually equals a.
• f(x): This is the function for which we are finding the limit.
• L: This represents the limit's value. It is the value that f(x) approaches as x approaches a.

Understanding Indeterminate Forms and Limits

Sometimes, directly substituting x = a into f(x) results in an indeterminate form, such as 0/0 or ∞/∞. In these cases, we use techniques like:

• Factoring: Simplifying the function by factoring and canceling common terms.
• Rationalizing: Multiplying the numerator and denominator by the conjugate.
• L'Hôpital's Rule: (For indeterminate forms of 0/0 or ∞/∞) Taking the derivative of the numerator and denominator separately and then evaluating the limit.
• Trigonometric Identities: Applying trigonometric identities to simplify the expression.

Example

Let's say we want to find the limit of the function f(x) = (x² - 1) / (x - 1) as x approaches 1.

lim_x→1 (x² - 1) / (x - 1)

If we directly substitute x = 1, we get (1² - 1) / (1 - 1) = 0/0, which is an indeterminate form.

To solve this, we can factor the numerator:

lim_x→1 ((x + 1)(x - 1)) / (x - 1)

Now, we can cancel the (x - 1) terms:

lim_x→1 (x + 1)

Now, we can substitute x = 1:

1 + 1 = 2

Therefore, lim_x→1 (x² - 1) / (x - 1) = 2

In conclusion, the limit formula describes how a function behaves as its input gets arbitrarily close to a specific value, and various techniques are used to evaluate limits, especially when direct substitution results in indeterminate forms.