(a) Use numerical and graphical evidence to guess the value...

(a) Use numerical and graphical evidence to guess the value of the limit
$$ \lim_{x \to 1}\frac{x^3 - 1}{\sqrt{x} - 1} $$

(b) How close to 1 does $ x $ have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?

Key Concepts

Limit of a Function

This concept involves understanding the behavior of a function as its input approaches a specific point. It is a fundamental idea in calculus that describes how a function 'settles' towards a particular value, even if it may not be defined exactly at that point. By investigating limits, one can infer local properties of the function and predict its behavior in the neighborhood of the target point.

Indeterminate Forms and Algebraic Simplification

When evaluating limits, one may encounter indeterminate expressions such as 0/0. These forms arise when both the numerator and the denominator approach zero as the input nears a certain value. Resolving an indeterminate form often requires algebraic techniques like factoring, rationalizing, or applying L’Hôpital’s Rule to simplify the expression and reveal the underlying limit.

Numerical and Graphical Analysis

Numerical evaluation and graphical methods serve as practical approaches to estimate and understand the behavior of functions near a point of interest. By computing function values close to the limit point and plotting the function, one can observe trends and approximate the limit, which is especially useful when analytical methods are intricate or when providing empirical evidence for theoretical findings.

Epsilon-Delta (?-?) Definition of a Limit

This is the formal mathematical definition of a limit, where for every positive tolerance level (epsilon), there exists a corresponding distance (delta) such that if the input is within delta of the target point, then the function’s value is within epsilon of the limit. This precise formulation is crucial for establishing the rigor behind limit statements and for quantifying how close the input must be to guarantee a specific proximity of the function to its limit.

Transcript

00:01 So in this problem, we're asked to use numerical and graphical methods to show what the limit is x goes to 1 of this function, x cubed minus 1 over the square root of x minus 1 equals.

00:30 So first of all, numerically, let's make a table here of x's and f of x.

00:41 Okay.

00:43 We'll have our xs go to 1, first from the left.

00:46 So if you put 0 .5 into this formula, you get 2 .987, 0 .75 in here.

00:56 You get 4 .315.

01:00 0 .90 in here.

01:02 You get 5 .281, inching our way towards 1, right? 0 .93 gives me 5 .49 .0 .99.

01:21 Gives me 5 .925.

01:26 0 .999.

01:30 Gives me 5 .9925.

01:33 All right, so this suggests that the limit, as x approaches one from the left of our function is six doesn't it all right now let's go from the other way let's start at 1 .1 that's 6 .782 and 1 .065 is 6 .501 1 .01 is 6 .0701 is 6 .075 1 .01 is 6 .075 .1 is 6 .075 1 .001.

02:21 See, i'm getting closer to 1.

02:23 6 .0075.

02:27 So this suggests that the limit as x approaches 1 from the right then, right? i'm coming in closer from the right of our function is also 6.

02:40 So, since i have the same limit from the left and the right, then i have the limit.

02:48 It's not writing very neatly.

02:50 Let me try that again.

02:52 Limit as x approaches 1 of x cubed minus 1 over the square root of x minus 1.

03:02 This function is 6, numerically, isn't it? all right, let's see it graphically now.

03:11 I look at it graphically...

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