Let f ( x ) = x 2 − 4 x − 2 f\left(x\right)=\frac{x^2-4}{x-2} f ( x ) = x − 2 x 2 − 4 . <IMAGE> Make a conjecture about the value of lim x → 2 x 2 − 4 x − 2 {\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}} x → 2 lim x − 2 x 2 − 4 .

Welcome back everyone using the given table of values for F FX equals X squared minus 25 divided by X minus five. What is the plausible value of the limit of X squared minus 25 divided by X minus five as X approaches five here, we have a table of X and F of X values. And for our answer choices, A says the limit is 10 B 25 C zero and D nine. Now, what's going on in our table here? Notice that it has X values, OK? And F FX values and for our X and F FX values in the top row for our X values notice that it's going from 4.9 to 4.99 to 4.999 and so on. So it's approaching five. Well, for our values in the third row of our table, the X values, OK? Or the second set of X values, it's 5.15 0.00 sorry 5.15 0.015 0.001 and so on. So it's approaching five from the right. So that means then that our FFX values will tell us the limit if they are both approaching the same value. Now, for F FX values, it's going from 9.9 to 9.99 to 9.999. And so on, notice that from our left hand side, it's approaching 10. If we look at it from the right hand side, it's going from 10.1 to 10.01 to 10.001 to 10.0001. So from the right hand side, we observe that it's also approaching 10 from the right. Therefore, it follows that the plausible value of the limit of F FX as X approaches five is 10 A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

2:03 minutes

Problem 7b

Textbook Question

Let $f\left(x\right)=\frac{x^2-4}{x-2}$ . <IMAGE>
Make a conjecture about the value of ${\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}}$ .

Verified step by step guidance

First, recognize that the expression $ \frac{x^2 - 4}{x - 2} $ is undefined at $ x = 2 $ because it results in division by zero. Therefore, we need to simplify the expression to evaluate the limit.

Notice that the numerator $ x^2 - 4 $ can be factored as a difference of squares: $ x^2 - 4 = (x - 2)(x + 2) $.

Substitute the factored form into the original expression: $ \frac{(x - 2)(x + 2)}{x - 2} $.

Cancel the common factor $ (x - 2) $ in the numerator and the denominator, which simplifies the expression to $ x + 2 $, provided $ x \neq 2 $.

Now, evaluate the limit of the simplified expression as $ x \to 2 $: $ \lim_{x \to 2} (x + 2) $. This can be directly computed by substituting $ x = 2 $ into the expression, leading to the conjecture about the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points where it may not be explicitly defined, such as points of discontinuity. In this question, we are interested in the limit of the function as x approaches 2, which is crucial for evaluating the function's value at that point.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or expressions. In the given function, f(x) = (x^2 - 4)/(x - 2), the numerator can be factored as (x - 2)(x + 2). This simplification is essential for evaluating the limit, as it allows us to cancel out the common factor in the numerator and denominator, making the limit easier to compute.

Continuity

Continuity refers to a function being unbroken or having no gaps at a certain point. A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. In this case, understanding continuity is important because it helps us determine whether the limit we compute will yield a valid function value at x = 2, which is initially undefined in the original function.

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