What is the limit of given function , which is given in attachment file

To determine the limit of a function, we typically analyze its behavior as the input approaches a certain value. Since I can't view attachments, let's discuss a general approach to finding limits, and I can guide you through the process with an example.

Understanding Limits

Limits help us understand the behavior of functions as they approach a specific point. For instance, if we have a function \( f(x) \), we might be interested in what happens to \( f(x) \) as \( x \) approaches a value \( a \). This can be expressed mathematically as:

lim (x → a) f(x)

Common Techniques for Finding Limits

There are several techniques you can use to find limits, depending on the function's form:

• Direct Substitution: If \( f(a) \) is defined and finite, then the limit is simply \( f(a) \).
• Factoring: If direct substitution results in an indeterminate form like \( \frac{0}{0} \), try factoring the expression and simplifying it.
• Rationalization: For functions involving square roots, multiplying by the conjugate can help eliminate indeterminate forms.
• L'Hôpital's Rule: If you encounter \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), differentiate the numerator and denominator separately and then take the limit again.

Example: Finding a Limit

Let’s consider a specific example to illustrate these techniques. Suppose we want to find:

lim (x → 2) (x² - 4) / (x - 2)

1. **Direct Substitution**: Plugging in \( x = 2 \) gives us \( \frac{0}{0} \), which is indeterminate.

2. **Factoring**: We can factor the numerator:

(x² - 4) = (x - 2)(x + 2)

So, our limit becomes:

lim (x → 2) (x - 2)(x + 2) / (x - 2)

3. **Canceling**: We can cancel \( (x - 2) \) from the numerator and denominator (as long as \( x \neq 2 \)):

lim (x → 2) (x + 2)

4. **Direct Substitution Again**: Now, substituting \( x = 2 \) gives us:

2 + 2 = 4

Final Result

Thus, the limit of the function as \( x \) approaches 2 is 4. This example illustrates how to handle limits effectively, using various techniques based on the function's characteristics.

If you have a specific function in mind, feel free to describe it, and I can help you work through the limit step by step!