Verify that lim 3x-5/4x^2+9=1/25
To verify the given limit, we will factor the numerator and denominator and then simplify the expression.
We start by factoring the numerator and denominator:
Numerator:
3x - 5
Denominator:
4x^2 + 9
Next, we rewrite the expression as:
lim(x→∞) (3x - 5)/(4x^2 + 9)
Now, we can divide every term in the expression by x^2, as x^2 grows much faster than 3x or 5:
lim(x→∞) (3/x - 5/x^2)/(4 + 9/x^2)
As x approaches infinity, (3/x) and (5/x^2) both approach zero:
lim(x→∞) (0 - 0)/(4 + 9/x^2)
This simplifies to:
lim(x→∞) 0/4
Since the numerator is 0 and the denominator is a constant (4), the limit is equal to 0 divided by 4, which is 0:
lim(x→∞) 0/4 = 0
Therefore, the given limit is equal to 0, not 1/25.
The question is incomplete since it does not say what value the x approaches.
It would make sense if it said :
find Limit (3x-5)/(4x^2+9) as x --- 2
the answer to that would be 1/25
You are correct, my apologies for the confusion. Without specifying the value of x that the limit is approaching, we cannot determine the value of the limit.
If the question is "find the limit (3x-5)/(4x^2+9) as x approaches 2," then we can find the value of the limit. Let's calculate:
lim(x→2) (3x - 5)/(4x^2 + 9).
Substituting x = 2 into the expression, we get:
lim(x→2) (3(2) - 5)/(4(2)^2 + 9)
= lim(x→2) (6 - 5)/(16 + 9)
= (6 - 5)/(16 + 9)
= 1/25.
Therefore, if x approaches 2, the limit of (3x - 5)/(4x^2 + 9) is indeed 1/25.
To verify the limit:
lim (3x-5)/(4x^2+9) = 1/25
We need to find the value of x that satisfies the limit equation.
The first step is to simplify the equation.
lim (3x-5)/(4x^2+9) = 1/25
To simplify, let's factor the denominator:
lim (3x-5)/[(2x)^2 + 3^2] = 1/25
lim (3x-5)/(4x^2 + 9) = 1/25
Now, we can compare the denominator to the standard form of a difference of squares:
a^2 - b^2 = (a + b)(a - b)
Here, a is 2x and b is 3:
4x^2 + 9 = (2x)^2 + 3^2 = (2x + 3)(2x - 3)
Substituting this back into the equation, we get:
lim (3x-5)/[(2x + 3)(2x - 3)] = 1/25
Now, let's cancel out the common factors of (2x + 3) from the numerator and denominator:
lim (3x-5)/(2x - 3) = 1/25
To find the value of x that satisfies the limit, we can set the numerator and denominator equal to zero:
3x - 5 = 0
Solving for x, we get:
3x = 5
x = 5/3
However, this value of x does not satisfy the denominator (2x - 3) = 0.
So, the equation does not hold true for any specific value of x.
Therefore, the given equation does not verify that lim (3x-5)/(4x^2+9) = 1/25.