In pre-calculus, the concept of limits is used to describe the behavior of a function as its input approaches a certain value. Limits are crucial in understanding calculus, especially when dealing with rates of change and continuity.

Key Points:

• Limit: A limit is the value that a function approaches as the input approaches a specific value. It can help us understand what happens to a function when it gets closer to a point, even if the function doesn't actually reach that point.
• Notation: The limit of a function f(x) as x approaches a value a is written as lim (x → a) f(x) = L, where L is the value that the function approaches.

Example 1: Simple Limit

Let's look at the function f(x) = 2x. We want to find the limit of this function as x approaches 3.
As x gets closer and closer to 3, the value of f(x) (which is 2 times x) gets closer and closer to 2 × 3 = 6.
So, we can say: lim (x → 3) 2x = 6.

Example 2: Limit with a Fraction

Now, let's consider a different function: f(x) = (x² - 1) / (x - 1).
If we try to directly substitute x = 1 into this function, we get (1² - 1) / (1 - 1) = 0 / 0, which is undefined (this is called an indeterminate form).
However, we can factor the numerator and simplify the expression:
(x² - 1) = (x - 1)(x + 1), so the function becomes:
f(x) = (x - 1)(x + 1) / (x - 1).
Now, we can cancel out the (x - 1) terms (for values of x ≠ 1), leaving:
f(x) = x + 1.
Now, we can safely substitute x = 1 into the simplified function:
f(1) = 1 + 1 = 2.
So, the limit of the function as x approaches 1 is: lim (x → 1) (x² - 1) / (x - 1) = 2.

Key Takeaways:

• Limits help us understand the behavior of functions as they approach certain values.
• Limits can be used to find the value of a function at a point where it may not be directly defined (like in the case of indeterminate forms).
• Limits are foundational in calculus, especially for understanding derivatives and integrals.

Summary:

A limit describes the value that a function approaches as the input gets closer to a specific value. Even if a function is not defined at that exact point, the limit helps us understand its behavior near that point. Limits are essential for understanding concepts in calculus such as continuity, derivatives, and integrals.

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