Question:
If lim(f(x)/x)=-5 as x approaches 0, then
lim(x^2(f(-1/x^2))) as x approaches infinity is equal to
(a) 5 (b) -5 (c) -infinity (d) 1/5 (e) none of these
The answer key says (a) 5.
So this is what I know:
Since lim(f(x)/x)=-5 as x approaches 0, then
lim(f(x))=-5x as x approaches 0
However, how could this information be used to evaluate the other limit since the other limit is approaching infinity?
If I were to just ignore this issue this is what I come on with:
Since f(x)=-5x then,
lim(f(-1/x^2))=-5(-1/x^2) as x approaches infinity
So I could simplify lim(x^2(f(-1/x^2))) as x approaches infinity to
lim(x^2(-5(-1/x^2))) as x approaches infinity, then cancel out the x^2 on the numerator and denominator and simplify. Then I'm left with
lim(5) as x approaches infinity, which is equal to 5.
However to reiterate my problem is how can I transfer the knowledge I've obtained from the first limit to the second limit since the first limit is approaching 0 and the second limit is approaching infinity?
Thank you in advance for the help.
let z = -1/x^2
as z->0, x->∞
lim(f(x)/x)=-5
lim(f(z)/z) = -5 as z->0
so, lim(f(-1/x^2)/(-1/x^2)) = - lim(-x^2f(-1/x^2)) = 5 as x->∞
To evaluate the second limit, we can use the information from the first limit and apply some algebraic manipulation.
Let's start with the first limit: lim(f(x)/x) = -5 as x approaches 0.
This tells us that as x approaches 0, the ratio of f(x) to x approaches -5. We can express f(x) as f(x) = -5x + g(x), where g(x) is a function that approaches 0 as x approaches 0.
Now, let's focus on the second limit: lim(x^2(f(-1/x^2))) as x approaches infinity.
We need to simplify the expression inside the limit. Using the expression for f(x) from above, we have:
lim(x^2(-5(-1/x^2))) as x approaches infinity.
Simplifying further, we get:
lim(5(x^2/x^2)) as x approaches infinity.
Canceling out the x^2 on the numerator and denominator, we have:
lim(5) as x approaches infinity.
The limit of a constant value is equal to the constant value. Therefore, the result is 5.
So, the correct answer is (a) 5.
In summary, to transfer the knowledge from the first limit to the second limit, we express f(x) in terms of x with an additional function that approaches 0, and then substitute this expression into the second limit.
To evaluate lim(x^2(f(-1/x^2))) as x approaches infinity, you can employ a change of variables to relate it to the given limit lim(f(x)/x) as x approaches 0.
Let's make the substitution u = -1/x^2. As x approaches infinity, u approaches 0. We can rewrite the limit as:
lim(x^2(f(-1/x^2))) as x approaches infinity
= lim(x^2(f(u))) as x approaches infinity
As x approaches infinity, u approaches 0. So, we can rewrite the limit in terms of u:
lim(x^2(f(u))) as x approaches infinity
= lim((-1/u) * (1/u) * f(u)) as u approaches 0
Now, let's substitute the given limit lim(f(x)/x)=-5 as x approaches 0:
lim((-1/u) * (1/u) * f(u)) as u approaches 0
= lim(-1/u * (1/u) * (-5u)) as u approaches 0
= lim(5/u^2) as u approaches 0
As u approaches 0, 1/u^2 approaches infinity. Therefore, the limit becomes:
lim(5/u^2) as u approaches 0
= 5 * lim(1/u^2) as u approaches 0
= 5 * infinity
= infinity
Hence, the correct answer is (e) none of these, as the limit is infinity, not 5.