(a) Use a graph of

f(x) = sqrt(2x^2+9x+5)−sqrt(2x^2+2x+1)

to estimate the value of
lim x→∞ f(x)
to one decimal place.

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

(c)Find the exact value of the limit.

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To estimate the value of the limit lim x→∞ f(x) using a graph, follow these steps:

(a) Use a graph:
1. Graph the function f(x) = sqrt(2x^2+9x+5)−sqrt(2x^2+2x+1).
2. Observe the behavior of the graph as x approaches positive infinity (∞).
3. Identify the y-value (f(x)) as x approaches a very large positive value.
4. Round the y-value to one decimal place to estimate the limit.

To estimate the value of the limit using a table of values:

(b) Use a table:
1. Create a table of x and f(x) values by plugging in various large positive values of x into the function f(x).
2. Observe how the f(x) values change as x becomes larger and larger.
3. Identify the y-value (f(x)) as x approaches a very large positive value.
4. Round the y-value to four decimal places to estimate the limit.

To find the exact value of the limit:

(c) Find the exact value:
1. Start by simplifying the function f(x):
f(x) = sqrt(2x^2+9x+5)−sqrt(2x^2+2x+1)
2. Rearrange the function to find a common denominator to simplify further:
f(x) = [sqrt(2x^2+9x+5)−sqrt(2x^2+2x+1)] * [(sqrt(2x^2+9x+5)+sqrt(2x^2+2x+1)) / (sqrt(2x^2+9x+5)+sqrt(2x^2+2x+1))]
3. Apply the difference of squares formula to eliminate square roots:
f(x) = [2x^2+9x+5 - (2x^2+2x+1)] / (sqrt(2x^2+9x+5)+sqrt(2x^2+2x+1))
f(x) = (7x+4) / (sqrt(2x^2+9x+5)+sqrt(2x^2+2x+1))
4. Observe that as x approaches infinity, both the numerator and denominator grow indefinitely.
5. Apply the limit properties: as the numerator and denominator approach infinity, the limit is of the form ∞/∞.
6. Use L'Hopital's rule to find the limit of the derivative of the numerator and denominator:
lim x→∞ f(x) = lim x→∞ (7) / (1 / (4 * (2x^2+9x+5)^0.5 + 1 / ((2x^2+2x+1)^0.5)
7. Simplify the expression:
lim x→∞ f(x) = 7 / (1 / (∞) + 1 / (∞))
lim x→∞ f(x) = 7 / (0 + 0)
8. Simplify further:
lim x→∞ f(x) = 7 / 0
9. Since dividing by zero is undefined, the exact value of the limit does not exist.

In summary, the steps provided guide you on estimating the value of the limit using a graph and a table. Additionally, by simplifying the function using algebraic manipulation and applying the concept of limits, you can find the exact value of the limit, if it exists.