Question

In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow 3} \frac{x-2}{x^{2}} $$

Short answer

The limit of the function \( \frac{x - 2}{x^2} \) as \( x \) approaches 3, is \( \frac{1}{9} \).

Step-by-step solution

Determine the Function

Our function is \( \frac{x - 2}{x^2} \) and we want to find the limit as x approaches 3.

Substitute the Value

Substitute x = 3 into our function: \( \frac{3 - 2}{3^2} = \frac{1}{9} \).

Simplify the expression

The expression \( \frac{1}{9} \) cannot be simplified further, hence this is our limit.

Key concepts

Limits and Continuity

Grasping the concept of limits is crucial for understanding the foundations of calculus. A limit describes the behavior of a function as its input approaches a certain value, but doesn't necessarily include that value. Imagine inching closer and closer towards a point without actually touching it. That's what limits in calculus are about — examining the tendency of functions near specific points.

In our exercise, we explored the limit as the variable 'x' approaches 3 for the function \( \frac{x - 2}{x^2} \). Continuity, on the other hand, means that at the point where we are finding the limit, the function is defined, and the limit equals the function's value at that point. In this case, the function is continuous at x = 3, because the value we obtained from the limit was exactly the value of the function at x = 3; there were no gaps, jumps, or holes in the graph of the function at this point.

Limit of a Function

The limit of a function is a fundamental concept in calculus, which helps in understanding the behavior of functions as they approach a particular value. In simple terms, it's like predicting the future value of a function without actually reaching that point in the present.

Let's look at our example: \( \lim_{{x \to 3}} \frac{x-2}{x^2} \). To find the limit, we substitute the approaching value into the function, assuming the function is defined and continuous at that point. The substitution gives us \( \frac{1}{9} \) as the limit. If substitution had led to an undefined value, we would have had to use other methods to find the limit, such as factoring, rationalizing, or applying special limit rules, like L'Hôpital's Rule.

Calculus Exercises

Engaging with calculus exercises is the best way to understand and master the various techniques and applications of calculus. From derivatives to integrals, and indeed limits, practice helps to reinforce theoretical knowledge and improve problem-solving skills. When dealing with limit problems, it's important to identify the type of function you're working with and determine the best strategy to approach the limit.

For example, in the given exercise, we find that direct substitution is our best approach. When facing more challenging limits, exploring graphical or numerical approaches can also become necessary. Remember, the key is to practice regularly and not to avoid tough problems—they often yield the greatest learning opportunities.