Understanding Limits: A Beginner's Guide with Examples

What Are Limits?

In calculus, a limit helps us understand the behavior of a function as it approaches a particular point. It’s like looking at the value a function gets closer to, without necessarily reaching that value.

Imagine you’re driving a car towards a stop sign. You may not come to a complete stop immediately, but as you approach the sign, your speed decreases, and your position gets closer and closer to the sign. This is similar to how limits work in math!

Example 1: Basic Limit Calculation

Let’s start with a simple function:

Function:

f(x) = 2x + 3

Finding the Limit:

We want to find the limit of f(x) as x approaches 4.

Step 1: Substitute x with 4 in the function:

f(4) = 2(4) + 3 = 8 + 3 = 11

Step 2: Conclusion:

So,

Limit:

\[ ext{lim}_{x \to 4} (2x + 3) = 11 \]

This means as x gets closer to 4, the value of f(x) approaches 11.

Example 2: Limit with a Fraction

Now, let’s look at a function that involves a fraction:

Function:

g(x) = (x² - 1) / (x - 1)

Finding the Limit:

We want to find the limit of g(x) as x approaches 1.

Step 1: Direct substitution gives us an undefined form:

g(1) = (1² - 1) / (1 - 1) = 0/0 (undefined)

Step 2: Factor the numerator:

g(x) = (x - 1)(x + 1) / (x - 1)

Step 3: Cancel out (x - 1):

g(x) = x + 1 (for x ≠ 1)

Step 4: Now substitute x with 1:

g(1) = 1 + 1 = 2

Step 5: Conclusion:

So,

Limit:

\[ ext{lim}_{x \to 1} \frac{x² - 1}{x - 1} = 2 \]

This means as x approaches 1, the value of g(x) approaches 2.

Example 3: One-Sided Limits

Sometimes we are interested in one-sided limits, which look at the behavior of a function from only one side (left or right).

Function:

h(x) = { 1, if x < 1; 2, if x ≥ 1 }

Finding the Left-Hand Limit (as x approaches 1 from the left):

Step 1: Check values just less than 1:

h(0.9) = 1

Step 2: Conclusion:

\[ ext{lim}_{x \to 1^-} h(x) = 1 \]

Finding the Right-Hand Limit (as x approaches 1 from the right):

Step 1: Check values just greater than 1:

h(1.1) = 2

Step 2: Conclusion:

\[ ext{lim}_{x \to 1^+} h(x) = 2 \]

Summary

Limits are a foundational concept in calculus that help us understand how functions behave. Whether you’re calculating simple limits or exploring more complex functions, remember: it’s all about what value the function approaches. Keep practicing with different functions, and soon enough, you’ll feel more comfortable with limits! Happy learning!