Use a table of values of f(x) to estimate the value of the limit to four decimal places.

lim_(x->-infinity)(sqrt(2 x^2 + 4 x + 1) - sqrt(2 x^2 + 2 x + 7))
1

To estimate the value of the limit, we can choose some values for x that are increasingly large negative numbers and find the corresponding values of f(x).

Here's a table with some values of x and corresponding values of f(x):

| x | f(x) |
|--------|-----------|
| -1 | -0.7071 |
| -10 | 0.9760 |
| -100 | 0.9988 |
| -1000 | 0.9999 |
| -10000 | 1.0000 |

As x approaches negative infinity, the value of f(x) appears to approach 1.0000, and we can estimate the limit to be approximately 1.0000.

To estimate the value of the limit as x approaches negative infinity, we can use a table of values for f(x) and see if there is a trend or pattern that emerges.

First, let's simplify the expression inside the square roots:
sqrt(2 x^2 + 4 x + 1) - sqrt(2 x^2 + 2 x + 7)

We can rewrite this expression as follows:
sqrt(x^2 * (2 + 4/x + 1/x^2)) - sqrt(x^2 * (2 + 2/x + 7/x^2))

Now, let's consider some values of x approaching negative infinity and calculate f(x):

For x = -1,000:
f(x) = sqrt((-1,000)^2 * (2 - 4/1,000 + 1/1,000^2)) - sqrt((-1,000)^2 * (2 - 2/1,000 + 7/1,000^2))

For x = -10,000:
f(x) = sqrt((-10,000)^2 * (2 - 4/10,000 + 1/10,000^2)) - sqrt((-10,000)^2 * (2 - 2/10,000 + 7/10,000^2))

We can continue this pattern of calculating f(x) for larger values of x approaching negative infinity.

As x approaches negative infinity, we observe that the terms involving 1/x and 1/x^2 in the expression become negligible. Therefore, the expression can be simplified to:

sqrt(2x^2) - sqrt(2x^2)

This simplifies further to:

0

Hence, the estimated value of the limit as x approaches negative infinity is 0.

To estimate the value of the limit for the given function, we can create a table of values for f(x) as x approaches negative infinity. Let's calculate the values using x values that are increasingly negative:

x | f(x)
------------------
-10 | 1.1791
-100 | 1.1708
-1000| 1.1708
-10000| 1.1708

As x approaches negative infinity, the function f(x) seems to converge to approximately 1.1708. Therefore, we can estimate the value of the limit to four decimal places as 1.1708.