Find the limit: lim x-> 2 ln(x/2)/ (x^2−4)
Can someone help me with this?
I thought that it was 1/4 but that's now the answer....
What I thought that I had to do was to take the derivative, (2/x)/2x and then plug in 2... and that's how I got 1/4, but that's not the answer, what do I have to do?
D'hôpital's rule works here.
The expression is
ln(x/2)/ (x^2−4)
not
2ln(x/2)/ (x^2−4)
as it appears in the post.
The leading 2 belongs to the limit of x.
Differentiate both top and bottom with respect to x:
Lim (1/x) / (2x)
= lim 1/(2x²)
=1/(2(2)²)
=1/8
To find the limit as x approaches 2 of ln(x/2) / (x^2−4), we can use a combination of algebraic manipulation and known limit properties.
Let's start by simplifying the expression. We have:
ln(x/2) / (x^2−4)
Recall the logarithmic property that ln(a/b) = ln(a) - ln(b). Applying this property, we can rewrite the expression as:
ln(x) - ln(2) / (x^2−4)
Now, let's factor the denominator:
x^2 - 4 = (x + 2)(x - 2)
Now, we can rewrite the expression as:
ln(x) - ln(2) / ((x + 2)(x - 2))
Next, we will use the fact that ln(2) is a constant, so the limit of ln(2) as x approaches 2 is ln(2).
Now, let's analyze the limit term-by-term:
- For ln(x), as x approaches 2, ln(x) approaches ln(2). This is because ln(x) is a continuous function at x = 2, and ln(2) is the natural logarithm of 2.
- For (x + 2)(x - 2), we can factor out the common factor (x - 2) and simplify the expression to (x - 2)(x + 2) = (x^2 - 4). As x approaches 2, (x^2 - 4) approaches (2^2 - 4) = 0.
- Finally, we have ln(2) / 0, which is undefined.
Since we end up with an undefined expression, it indicates that the limit as x approaches 2 of ln(x/2) / (x^2−4) does not exist.
In summary, the limit of ln(x/2) / (x^2−4) as x approaches 2 is undefined.