F(x)=(2x-1)/ (|x| -3)
Rewrite f(x) as a piecewise function. Then find the limit as it approaches positive infinity and negative infinity.
for x<0, |x| = -x. So, we have
f(x) =
(2x-1)/(-x-3) for x<0
(2x-1)/(x-3) for x>=0.
Now you can easily find the limits, no?
To confirm your answer, see
http://www.wolframalpha.com/input/?i=%282x-1%29%2F+%28|x|+-3%29
and scroll down to where it describes the limits.
Sure, let's give it a go!
To rewrite f(x) as a piecewise function, let's consider the two possible cases for x:
1) For x > 3:
In this case, the denominator |x| - 3 will always be positive, so we can rewrite f(x) as follows:
f(x) = (2x - 1)/(x - 3)
2) For x < -3:
In this case, the denominator |x| - 3 will also always be positive, but now the value of x - 3 will be negative. So, we can rewrite f(x) as follows:
f(x) = (2x - 1)/(-x - 3)
Now, let's compute the limit as x approaches positive infinity:
lim(x→∞) [ (2x - 1)/(x - 3) ]
Well, as x becomes infinitely large, the 1 and 3 in the numerator and denominator become insignificant. So, we can roughly say that the limit is:
lim(x→∞) (2x/x) = 2
Haha, looks like f(x) can't escape the influence of the number 2, even as it approaches infinity!
Next, let's compute the limit as x approaches negative infinity:
lim(x→-∞) [ (2x - 1)/(-x - 3) ]
Similarly, as x becomes infinitely negative, the 1 and 3 in the numerator and denominator become insignificant. So, we can roughly say that the limit is:
lim(x→-∞) (-2x/x) = -2
Haha, f(x) just can't get enough of the number 2, even when it heads towards negative infinity!
Hope that puts a smile on your face! If you have any more questions, feel free to ask!
To rewrite the function f(x) = (2x - 1) / (|x| - 3) as a piecewise function, we need to consider the behavior of the function around the critical point x = 3, where the denominator becomes zero.
For x < -3:
In this interval, |x| - 3 is negative while (2x - 1) is positive. Therefore, we can write f(x) as (2x - 1) / (|x| - 3) = (2x - 1) / (-x - 3).
For -3 < x < 3:
In this interval, |x| - 3 is negative while (2x - 1) is negative. Therefore, we can write f(x) as (2x - 1) / (|x| - 3) = (2x - 1) / (-x - 3).
For x > 3:
In this interval, |x| - 3 is positive while (2x - 1) is positive. Therefore, we can write f(x) as (2x - 1) / (|x| - 3) = (2x - 1) / (x - 3).
Now, let's find the limit of f(x) as x approaches positive infinity:
lim(x → ∞) f(x) = lim(x → ∞) (2x - 1) / (|x| - 3)
As x approaches positive infinity, both numerator and denominator grow without bounds. Therefore, we can use the highest degree terms in the numerator and denominator to determine the limit.
lim(x → ∞) (2x - 1) / (|x| - 3) = lim(x → ∞) (2x / x) = lim(x → ∞) 2 = 2
The limit of f(x) as x approaches positive infinity is 2.
Next, let's find the limit of f(x) as x approaches negative infinity:
lim(x → -∞) f(x) = lim(x → -∞) (2x - 1) / (|x| - 3)
As x approaches negative infinity, both numerator and denominator grow without bounds again. Therefore, we can use the highest degree terms in the numerator and denominator to determine the limit.
lim(x → -∞) (2x - 1) / (|x| - 3) = lim(x → -∞) (- 2x / -x) = lim(x → -∞) 2 = 2
The limit of f(x) as x approaches negative infinity is also 2.
To rewrite the function f(x) as a piecewise function, we need to consider the different cases for x.
Case 1: x > 0
In this case, the expression |x| in the denominator of the original function can be simplified to just x. So, f(x) becomes:
f(x) = (2x - 1) / (x - 3)
Case 2: x < 0
In this case, the expression |x| in the denominator of the original function can be simplified to -x. Also, the numerator (2x - 1) becomes -(2x - 1) as we multiply it by -1. So, f(x) becomes:
f(x) = -(2x - 1) / (-x - 3)
= (1 - 2x) / (x + 3)
Now, let's find the limit as x approaches positive infinity:
lim(x->∞) f(x) = lim(x->∞) [(2x - 1) / (x - 3)]
To determine the limit, we can compare the degrees of the numerator and denominator. The degree of the numerator is 1 (since the highest power of x is x^1), and the degree of the denominator is also 1. In this case, we take the ratio of the coefficients of the highest power of x, which is 2. Therefore, the limit as x approaches positive infinity is 2.
Next, let's find the limit as x approaches negative infinity:
lim(x->-∞) f(x) = lim(x->-∞) [(1 - 2x) / (x + 3)]
Again, comparing the degrees of the numerator and denominator, both are 1. So, we take the ratio of the coefficients of the highest power of x, which is -2. Therefore, the limit as x approaches negative infinity is -2.
So, the rewritten piecewise function is:
f(x) =
{
(2x - 1) / (x - 3) if x > 0,
(1 - 2x) / (x + 3) if x < 0
}
The limit as x approaches positive infinity is 2, and the limit as x approaches negative infinity is -2.