Assuming a,b,c and d are positive constants use algebra to find lim (ax^+b)/(cx+d)
x-->infinity
I am not certain what you mean by x^. Is that x squared? or x to the b power?
It's x squared.
lim (ax^2+b)/(cx+d)
x-->infinity
To find the limit of (ax^2 + b)/(cx + d) as x approaches infinity, we can use algebraic manipulation.
First, let's focus on the leading terms of the numerator and denominator, as these will dominate the behavior of the expression as x approaches infinity.
As x approaches infinity, the term ax^2 in the numerator and the term cx in the denominator become the largest terms. We can ignore all other terms because they become insignificant compared to these dominant terms.
Now, divide every term in the numerator and denominator by x^2, which will cancel out the x^2 term in the numerator and cx term in the denominator:
(ax^2 + b)/(cx + d) = (a(x^2/x^2) + b/x^2)/(c(x/x) + d/x) = (a + b/x^2)/(c + d/x)
As x approaches infinity, 1/x^2 becomes very small, and 1/x becomes close to 0. Therefore, the expression can be simplified further:
lim (ax^2 + b)/(cx + d) as x approaches infinity = lim (a + b/x^2)/(c + d/x) as x approaches infinity
Since both the numerator and denominator approach constants (a and c, respectively), as x approaches infinity, the limit becomes:
lim (ax^2 + b)/(cx + d) as x approaches infinity = a/c
Therefore, the limit of (ax^2 + b)/(cx + d) as x approaches infinity is equal to a/c, where a, b, c, and d are positive constants.