solve the limits question given in attachment.....................................................................................

It seems like there was a mention of an attachment, but I can't view attachments directly. However, I can certainly help you understand how to solve limits in calculus! Let’s go through some common techniques and examples that can help you tackle limits effectively.

Understanding Limits in Calculus

Limits are fundamental in calculus, helping us understand the behavior of functions as they approach a certain point. They are essential for defining derivatives and integrals. When we talk about the limit of a function as it approaches a specific value, we are interested in what value the function gets closer to, even if it never actually reaches that value.

Basic Limit Concepts

To grasp limits, consider the following key ideas:

• Approaching a Value: Limits examine what happens to a function as the input approaches a certain number.
• Left-Hand and Right-Hand Limits: These are the values that the function approaches from the left side (negative direction) and the right side (positive direction).
• Existence of Limits: A limit exists only if both the left-hand and right-hand limits are equal.

Common Techniques for Solving Limits

Here are some strategies you can use to solve limit problems:

1. Direct Substitution

Start by substituting the value into the function. If you get a real number, that’s your limit. For example, if you want to find the limit of f(x) = 2x + 3 as x approaches 1:

f(1) = 2(1) + 3 = 5. Thus, the limit is 5.

2. Factoring

If direct substitution results in an indeterminate form like 0/0, try factoring. For instance, consider the limit of (x² - 1)/(x - 1) as x approaches 1:

Factor the numerator: (x - 1)(x + 1)/(x - 1). Cancel the (x - 1) terms, and you get the limit of x + 1 as x approaches 1, which is 2.

3. Rationalization

For limits involving square roots, rationalizing can help. For example, to find the limit of (√x - 2)/(x - 4) as x approaches 4:

Multiply the numerator and denominator by the conjugate (√x + 2). This simplifies the expression, allowing you to evaluate the limit more easily.

4. L'Hôpital's Rule

If you encounter an indeterminate form like 0/0 or ∞/∞, L'Hôpital's Rule can be applied. Differentiate the numerator and denominator separately, then take the limit again. For example, for lim (x→0) (sin x)/x, both the numerator and denominator approach 0. Applying L'Hôpital's Rule gives you lim (x→0) (cos x)/1 = 1.

Example Problem

Let’s say you want to find the limit of (3x² - 12)/(x - 4) as x approaches 4:

• First, substitute x = 4: (3(4)² - 12)/(4 - 4) = (48 - 12)/(0) = 36/0, which is undefined.
• Next, factor the numerator: 3(x² - 4) = 3(x - 2)(x + 2).
• Now, rewrite the limit: lim (x→4) [3(x - 2)(x + 2)]/(x - 4). This still leads to an indeterminate form.
• Instead, we can analyze the behavior as x approaches 4. The function approaches infinity or negative infinity, depending on the direction from which x approaches 4.

Final Thoughts

Limits can be tricky, but with practice, you’ll become more comfortable with them. Remember to explore different techniques based on the problem at hand. If you have a specific limit question in mind, feel free to describe it, and I can guide you through the solution step by step!