Can someone help me with the problem?
f(x)={ln 2x, 0<x<2
{2lnx, x >or= 2
The limit lim x-> 2 f(x) is?
is it 2ln 2?
f(x) => ln 4 when x is "barely less than" 2 and then becomes 2 ln2 at x = 2 These two numbers are equal. The funtion has a slope discontinuity at x=2 but remains continuous there.
The limit is 2ln2 = ln4 = 1.3863..
o alright thank you drwls
By definition of Limit :
The left limit value should be equal to right limit value .
lim f(x) = lim f(x) = lim f(x)
x->a- x->a+ x->a
In our problem say
The left limit values lies between 0 to 2
lim ln 2x = lim 2 ln x
x->2- x->2+
==> ln 4 = 2 ln 2
[Hint ln a ^b = b ln a ]
[ln 4 = ln 2^2 = 2 ln 2 ]
==> ln 4 = ln 4
So the left limit should be equal to right limit.
So limit exist at x = 2
Thanks !
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To find the limit as x approaches 2 of the function f(x), we need to evaluate the function for values of x that are very close to 2 from both sides.
First, let's consider the left-hand limit as x approaches 2 (x < 2). In this case, the function f(x) is defined as ln(2x). To find the left-hand limit, we substitute 2 into the function, but we need to consider values of x that are less than 2. Let's choose a value that is slightly less than 2, such as 1.9:
lim x->2- f(x) = lim x->2-(ln 2x) = ln(2 * 1.9)
Next, let's consider the right-hand limit as x approaches 2 (x ≥ 2). In this case, the function f(x) is defined as 2ln(x). To find the right-hand limit, we substitute 2 into the function, but we need to consider values of x that are greater than or equal to 2. Let's choose a value that is slightly greater than 2, such as 2.1:
lim x->2+ f(x) = lim x->2+(2ln x) = 2ln(2.1)
Now, to find the overall limit as x approaches 2, we compare the left-hand and right-hand limits:
lim x->2- f(x) = ln(2 * 1.9) ≈ -0.089
lim x->2+ f(x) = 2ln(2.1) ≈ 0.410
Comparing these values, we see that the left-hand limit is approximately -0.089 and the right-hand limit is approximately 0.410. Since the left-hand and right-hand limits do not agree, the overall limit as x approaches 2 does not exist.
Therefore, the limit, lim x->2 f(x), does not exist for the given function f(x).