Limit of u(t)- u(t -\delta) as delta goes to zero, LTI systems

[AMac33]

Homework Statement

what is the limit of yb(t) as \delta goes to zero.

yb(t) = (b/delta)*exp[-t/RC](exp[delta/RC] - 1)(u(t)-u(t-delta))

b=1.






2. The attempt at a solution

I used L'hopitals rule to find the limit of (b/delta)*exp[-t/RC](exp[delta/RC] - 1), which i got to be exp[-t/RC]/RC.

But I do not know what to do with the u(t)-u(t-delta). does it go to zero? or am I supposed to say it goes to delta(t)?

Any help/ advice would be appreciated, thanks!

 

[CompuChip]

Does the expression

$$\lim_{\delta \to 0} \frac{u(t + \delta) - u(t)}{\delta}$$
ring a bell?

 

[AMac33]

honestly, no. I'm fairly new to the unit step and dirac delta function.

 

[CompuChip]

Oh, but this doesn't have to do with either of those. $\delta$ is just a variable, if it makes you feel any better you can call it x or (more commonly used) h

Maybe I should ask it the other way around: what is the definition of the derivative u'(t) of u(t) at t?

 

[DeeNos]

I would approach this problem by first understanding the terms and variables involved. In this case, we are dealing with a limit and an expression involving u(t) and u(t-delta) as delta goes to zero. From the given expression, we can see that b is a constant and RC is a parameter that affects the function. We also know that u(t) is the unit step function, which is equal to 1 for t >= 0 and 0 for t < 0.

Now, let's focus on the expression (u(t) - u(t-delta)). As delta goes to zero, this expression becomes 1 for all t, since u(t) is equal to 1 for all t >= 0. Therefore, we can rewrite the original expression as (b/delta)*exp[-t/RC](exp[delta/RC] - 1).

Using L'Hopital's rule, we can find the limit of this expression as delta goes to zero. This gives us exp[-t/RC]/RC. However, we cannot ignore the unit step function in the original expression. As we have established, as delta goes to zero, (u(t) - u(t-delta)) becomes 1 for all t. Therefore, the final limit of yb(t) would be 1/RC * exp[-t/RC] as delta goes to zero.

In conclusion, the limit of yb(t) as delta goes to zero is 1/RC * exp[-t/RC]. This result is valid for all values of b and RC, as long as they are not equal to zero. This is because the unit step function is only defined for non-negative values of t, and the exponential function is defined for all real numbers.