Find the lim x->infinite (x/[3x+5]) where [] denotes the greatest integer function use the squeeze theorem.
The greatest integer function of x is the equivalent of the floor function, and is the greatest integer that does not exceed x.
Since [x+3] is the floor function of x+3,
we have x+2≤[x+3]≤x+3
Let
L1=Lim x→∞ x/(x+2), and
L2=Lim x→∞ x/(x+3)
Evaluating, L1=1, L2=1, and therefore by the squeeze theorem (sandwich theorem),
lim x→∞ x/[x+3] = 1
To find the limit of the given function using the squeeze theorem, we need to find two other functions that bound the given function and whose limits we can easily calculate.
Let's start by finding the bounds using the greatest integer function ([]).
For any positive real number, x, the greatest integer function ([]), denoted as [x], returns the largest integer less than or equal to x. Since [3x + 5] is an integer function, we can rewrite it as:
[3x + 5] = 3x + 5 - {3x + 5}
Where {3x + 5} represents the fractional part of 3x + 5.
Now, let's consider the greatest integer function for the function f(x) = x/[3x + 5]:
[3x + 5] ≤ 3x + 5 [1]
0 ≤ {3x + 5} < 1 [2]
Dividing both sides of equation [1] by 3x + 5 gives:
[3x + 5] / (3x + 5) ≤ 1
Since [3x + 5] ≤ 3x + 5, we can rewrite equation [2] as:
0 ≤ {3x + 5} < [3x + 5] + 1
Now, divide both sides by (3x + 5):
0 / (3x + 5) ≤ {3x + 5} / (3x + 5) < ([3x + 5] + 1) / (3x + 5)
Simplifying further:
0 ≤ {3x + 5} / (3x + 5) < 1 [3]
Now, let's find the limits of the lower and upper bounds.
Lower Bound:
Taking the limit as x approaches infinity for equation [3], we get:
lim(x->∞) (0 / (3x + 5)) ≤ lim(x->∞) ({3x + 5} / (3x + 5)) < lim(x->∞) (1)
0 ≤ lim(x->∞) (1) ≤ 1
Thus, the lower bound limit is 0.
Upper Bound:
Taking the limit as x approaches infinity for equation [3], we get:
lim(x->∞) (1) ≤ lim(x->∞) ({3x + 5} / (3x + 5)) < lim(x->∞) (1)
1 ≤ lim(x->∞) ({3x + 5} / (3x + 5)) ≤ 1
Thus, the upper bound limit is also 1.
Since the lower bound is 0 and the upper bound is 1, we can conclude that the limit as x approaches infinity of the function f(x) = x/[3x + 5], using the squeeze theorem, is 1.
To find the limit of a function as x approaches infinity, we can use the Squeeze Theorem if we can find two other functions that "squeeze" the original function and have the same limit as x approaches infinity.
Let's start by finding the upper and lower bounds for the function f(x) = x/[3x+5].
For the upper bound, we know that the greatest integer function, [x], always rounds down the value of x to the nearest integer. So, we have:
f(x) = x/[3x+5] ≤ x/(3x) = 1/3,
since for any positive value of x, 3x is always greater than 5, and dividing by a larger number makes the fraction smaller.
For the lower bound, we can use the property that the greatest integer function rounds down and subtract 1 from x, giving:
f(x) = x/[3x+5] ≥ (x-1)/(3x+5).
Now, let's calculate the limit of the upper and lower bounds as x approaches infinity:
lim x->∞ (1/3) = 1/3, and
lim x->∞ [(x-1)/(3x+5)] = (∞-1)/3∞ = 1/3,
since the highest-order term (x) dominates the other terms as x approaches infinity.
Therefore, we have the following inequalities using the Squeeze Theorem:
1/3 ≤ f(x) ≤ 1/3.
Since the upper and lower bounds both have the same limit of 1/3 as x approaches infinity, by the Squeeze Theorem, the original function f(x) = x/[3x+5] also has a limit of 1/3 as x approaches infinity.
Hence, the lim x->∞ (x/[3x+5]) where [] denotes the greatest integer function is equal to 1/3.