kindly refer to the attachment. the question is from the chapter limits.

It seems like you're referring to a specific question from your limits chapter, but I can't view attachments. However, I can certainly help you understand concepts related to limits in calculus! If you can describe the question or the concept you're struggling with, I can provide a detailed explanation.

Understanding Limits in Calculus

Limits are fundamental in calculus, serving as the foundation for derivatives and integrals. Essentially, a limit helps us understand the behavior of a function as it approaches a certain point, whether that point is within the domain of the function or not.

The Concept of a Limit

When we talk about the limit of a function f(x) as x approaches a value 'a', we are interested in what value f(x) gets closer to as x gets closer to 'a'. This can be expressed mathematically as:

lim (x → a) f(x) = L

This notation means that as x approaches 'a', the function f(x) approaches the value L. It's important to note that the function does not necessarily need to equal L at x = a; it just needs to get arbitrarily close to L as x approaches a.

Types of Limits

• One-Sided Limits: These limits consider the behavior of the function as x approaches 'a' from one side only. The left-hand limit (approaching from the left) is denoted as lim (x → a-) f(x), and the right-hand limit (approaching from the right) is lim (x → a+) f(x).
• Infinite Limits: Sometimes, as x approaches a certain value, the function may increase or decrease without bound. In such cases, we say the limit is infinite, expressed as lim (x → a) f(x) = ∞ or -∞.
• Limits at Infinity: This refers to the behavior of a function as x approaches infinity or negative infinity. For example, lim (x → ∞) f(x) examines what happens to f(x) as x grows larger and larger.

Finding Limits: Techniques

There are several techniques to evaluate limits, including:

• Direct Substitution: If f(a) is defined, simply substitute 'a' into the function.
• Factoring: If direct substitution results in an indeterminate form like 0/0, try factoring the expression to simplify it.
• Rationalization: This is particularly useful for limits involving square roots. Multiply the numerator and denominator by the conjugate to eliminate the radical.
• L'Hôpital's Rule: If you encounter an indeterminate form (0/0 or ∞/∞), you can differentiate the numerator and denominator separately and then take the limit again.

Example of a Limit Calculation

Let’s consider the limit:

lim (x → 2) (x² - 4)/(x - 2)

If we substitute x = 2 directly, we get 0/0, an indeterminate form. To resolve this, we can factor the numerator:

(x² - 4) = (x - 2)(x + 2)

Now, the limit becomes:

lim (x → 2) [(x - 2)(x + 2)/(x - 2)]

We can cancel (x - 2) from the numerator and denominator (as long as x ≠ 2), leading to:

lim (x → 2) (x + 2) = 4

Thus, the limit is 4.

Practical Applications of Limits

Limits are not just theoretical; they have practical applications in various fields, including physics, engineering, and economics. For example, they help in determining instantaneous rates of change (derivatives) and in understanding the behavior of functions at boundaries.

Feel free to share more details about your specific question or concept, and I can provide further assistance tailored to your needs!