Find the limit of this calculus problem.?
(5x-3)/(ln(5+4e^(x))
x=inf
Intuitive approach
for large values of x, the 5 of (5 + e^x) becomes insignificant , so for all practical purposes
we just have to look at ln(4e^x)
which is ln4 + lne^x
= ln4 + x
Also for very large values of x, the constant ln4 is relatively insignificant, so the denominator
approaches x for large values of x
Now look at the top.
Again for large values of x , the -3 becomes meanignless compared to the size of 5x
So for large values of x
our expression becomes
lim 5x/x as x ----> infinity
= 5
5x+2xa-6ab
To find the limit of the given calculus problem as x approaches positive infinity, we need to apply the rules of limits. In this case, we will use the concept of the dominant term.
Let's break down the given expression:
(5x - 3)/(ln(5 + 4e^x))
As x approaches positive infinity, the exponential term e^x dominates the expression because it grows much faster than x.
So, we can simplify the expression by neglecting the smaller terms in comparison to e^x. We can ignore the constant term (-3) as it becomes negligible compared to e^x.
Now, the expression becomes:
(5x)/(ln(5 + 4e^x))
Next, we divide the numerator and denominator by e^x to eliminate the exponential term. Remember that dividing by e^x is the same as multiplying by e^(-x).
(5x * e^(-x))/(ln(5e^(-x) + 4))
Now, as x approaches positive infinity, e^(-x) approaches zero, and 5e^(-x) + 4 approaches 4. This is because the exponential term becomes very small as x becomes very large.
So, we can simplify the expression further:
(5x * e^(-x))/(ln(4))
Since the denominator is a constant (ln(4)) and the numerator contains x multiplied by e^(-x), we can evaluate the limit by separating it:
lim(x -> inf) (5x * e^(-x))/(ln(4)) = lim(x -> inf) (5x)/(ln(4)) * lim(x -> inf) e^(-x)
The limit of (5x)/(ln(4)) as x approaches infinity is infinity.
The limit of e^(-x) as x approaches infinity is 0.
Therefore, the overall limit is:
lim(x -> inf) (5x * e^(-x))/(ln(4)) = infinity * 0 = 0
Hence, the limit of the given calculus problem as x approaches positive infinity is 0.