A metal cable has radius r and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable is
v = -C(r/R)^2*ln(r/R)
where c is a positive constant. Find the following limits and interpret your answer
(a) lim R--->r+ (v)
(b) lim r----->0+ (v)
0 and 1
To find the limits as R approaches r+ and as r approaches 0+, we will substitute the given expression for v into the limits and simplify.
(a) lim R ---> r+ (v)
We substitute the expression for v:
lim R ---> r+ (-C(r/R)^2 * ln(r/R))
Now, let's simplify the expression inside the parentheses:
(r/R)^2 = r^2/R^2
Now, substitute the simplified expression back into the limit:
lim R ---> r+ (-C(r^2/R^2) * ln(r/R))
Since r and C are constants, let's rename -Cr^2 as another constant, let's say, A:
lim R ---> r+ (A/R^2 * ln(r/R))
Now, as R approaches r+, the term R^2 in the denominator becomes arbitrarily small. At the same time, ln(r/R) approaches -∞. Thus, the limit becomes:
lim R ---> r+ (A/R^2 * ln(r/R)) = 0 * (-∞) = 0
Therefore, the limit as R approaches r+ for v is 0. This means that as the insulation layer approaches zero thickness, the velocity of the electrical impulse inside the cable approaches zero.
(b) lim r ---> 0+ (v)
We substitute the expression for v:
lim r ---> 0+ (-C(r/R)^2 * ln(r/R))
Similarly, let's simplify the expression inside the parentheses:
(r/R)^2 = r^2/R^2
Now, substitute the simplified expression back into the limit:
lim r ---> 0+ (-C(r^2/R^2) * ln(r/R))
As r approaches 0, both r^2 and r/R approach 0. Additionally, ln(r/R) approaches -∞. Thus, the limit becomes:
lim r ---> 0+ (-C(0/0) * (-∞))
The expression (-C(0/0)) is an indeterminate form. We cannot directly determine the limit in this case. To evaluate this limit, we would need more information or specify the relationship between the variables r and R.
Therefore, the limit as r approaches 0+ for v is indeterminate and further information is needed to interpret the result.