Question Evaluate $\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$ using the provided table of $x$ and $f(x)$ values.

Studdy Solution

STEP 1

Assumptions
1. The table provides values of $f(x)$ for $x$ approaching $1$ from both the left and the right.
2. The function $f(x)$ is defined as $f(x) = \frac{\sqrt{x}-1}{x-1}$ .
3. The limit we want to evaluate is $\lim _{x \rightarrow 1} f(x)$ .

STEP 2

To evaluate the limit numerically, we should look at the values of $f(x)$ as $x$ gets closer and closer to $1$ from both sides.

STEP 3

Examine the values of $f(x)$ as $x$ approaches $1$ from the left (i.e., $x$ is less than $1$ ).

STEP 4

List the values of $f(x)$ as $x$ approaches $1$ from the left:
- For $x = 0.9$ , $f(x) = 0.513167019494862$ - For $x = 0.99$ , $f(x) = 0.501256289338003$ - For $x = 0.999$ , $f(x) = 0.500125062539047$ - For $x = 0.9999$ , $f(x) = 0.500012500624980$

STEP 5

Observe that as $x$ gets closer to $1$ from the left, the values of $f(x)$ are getting closer to $0.5$ .

STEP 6

Examine the values of $f(x)$ as $x$ approaches $1$ from the right (i.e., $x$ is greater than $1$ ).

STEP 7

List the values of $f(x)$ as $x$ approaches $1$ from the right:
- For $x = 1.1$ , $f(x) = 0.488088481701516$ - For $x = 1.01$ , $f(x) = 0.498756211208895$ - For $x = 1.001$ , $f(x) = 0.499875062461019$ - For $x = 1.0001$ , $f(x) = 0.499987500624021$

STEP 8

Observe that as $x$ gets closer to $1$ from the right, the values of $f(x)$ are also getting closer to $0.5$ .

STEP 9

Notice that the values of $f(x)$ from both sides are converging to the same number, $0.5$ , as $x$ approaches $1$ .

STEP 10

Based on the numerical evidence from the table, we can infer that the limit is $0.5$ .
$\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1} = 0.5$
This is a numerical approximation based on the given table of values. For a more rigorous approach, one would typically use algebraic manipulation or L'Hôpital's Rule to find the exact limit.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms