Find numbers a and b such that:
lim (sqrt(ax+b)-9)/x =1
x->0
lim (√(ax+b)-9)/x
= lim a/(2√(ax+b))
so, since we afre clearly dealing with lim = 0/0, we need
√(ax+b) = 9
ax+b = 81
a/(2√(ax+b)) = 1
a = 2√(ax+b)
a^2 = 4(ax+b)
a^2 = 4*81
a = 18
so, b=81
lim (√(18x+81)-9)/x = 1
verify at
https://www.wolframalpha.com/input/?i=limit+%28x-%3E0%29+%28%E2%88%9A%2818x%2B81%29-9%29%2Fx
To find the values of a and b such that the limit of the given expression approaches 1 as x approaches 0, we can use algebraic manipulation.
First, let's simplify the expression by rationalizing the numerator:
lim ((sqrt(ax+b)-9)/x) * ((sqrt(ax+b)+9)/(sqrt(ax+b)+9)) = 1
x->0
This gives us:
lim ((ax+b) - 81)/(x * (sqrt(ax+b) + 9)) = 1
x->0
Next, we can simplify further by multiplying both numerator and denominator by the conjugate of the denominator, which is (sqrt(ax+b) - 9):
lim ((ax+b) - 81) * (sqrt(ax+b) - 9)/(x * (sqrt(ax+b) + 9) * (sqrt(ax+b) - 9)) = 1
x->0
This simplifies to:
lim (ax + b - 81)/(x * (ax + b - 81)) = 1
x->0
Now we can cancel out the common factor in the numerator and denominator:
lim 1/(x) = 1
x->0
Therefore, to make the limit approach 1 as x approaches 0, we can choose values for a and b such that ax + b - 81 = 0. Solving this equation for a and b will give us the desired values.
Let's solve for b:
ax + b - 81 = 0
b = 81 - ax
Substitute this value of b back into the equation:
ax + (81 - ax) - 81 = 0
ax - ax = 0
Since the ax terms cancel out, we are left with 0 = 0. This means that any value of a and b will satisfy the equation.
In conclusion, there are infinitely many values for a and b that will make the limit of the given expression approach 1 as x approaches 0.
To find numbers a and b such that the given limit is equal to 1, we can use a method called L'Hospital's Rule.
L'Hospital's Rule states that if we have a limit in the form lim (f(x)/g(x)) as x approaches a, and both f(x) and g(x) approach 0 or infinity, then the limit of the ratio is equal to the limit of the ratio of the derivatives of f(x) and g(x).
Let's apply L'Hospital's Rule to the given limit.
First, let's rewrite the limit in a different form:
lim (sqrt(ax+b)-9)/x = 1
lim ((sqrt(ax+b)-9)/x) * x/x = 1
lim (sqrt(ax+b) - 9) / (x/x) = 1
lim (sqrt(ax+b) - 9) = 1 * 0
lim sqrt(ax+b) = 9
Now, let's square both sides to eliminate the square root:
lim (sqrt(ax+b))^2 = 9^2
lim (ax+b) = 81
Since we are taking the limit as x approaches 0, the value of ax + b should be equal to 81.
Therefore, we have the equation ax + b = 81.
To find the values of a and b, we need another equation to solve the system. However, the given limit does not provide any other information. Without additional constraints or information, it is not possible to determine the specific values of a and b.