The topic of this problem is Energy Storage Elements and Switched Circuits. The problem is defined Vc of t and t is equal to 0 minus and Vc of t and t is equal to infinity for the circuit shown below. So we have a circuit that has a DC voltage source in it, it has three resistors and a capacitor in it. We also have a switch in this circuit and the switch is closing at time t is equal to zero. So before time t is equal to zero, the switch is open and this path is an open circuit. After time t is equal to zero, it become for short circuit path. We know that our capacitor on the right hand side of our socket access an open socket in steady state for a DC voltage source or crime source. So in a DC circuit, capacitor which are in steady state act as an open circuit. So if we want to find Vc at t at t is equal to 0 minus, we'll evaluate the circuit before the switch is thrown and to find Vc at t at t is equal to infinity, we'll evaluate this circuit after the switch is been thrown in as closed to create a sort short circuit. So in order to do this, we first redraw our circuit for the two conditions. The first one is t is equal to 0 minus. What does that circuit look like before that switch closes? We have the voltage source in it, the sub s. We have our open circuit because the switch is not closed. We have our resistor. And we have our capacitor which is acting as an open circuit. And a DC steady state circuit. So we have R1, R2, R3, and our Vc of t is equal to 0 minus is measured across our open-circuited capacitor. So if we look at the circuit in this state, it becomes fairly easy to decide what Vc at t is equals 0 minus is. We know we can use voltage division to find Vc at t is equals to 0 minus. It's simply going to be the amount of the source voltage which is dropped across the R3 resistor at the right hand side of our circuit. So Vc at t is equal to 0 minus is going to be V sub s times R3 divided by R1 plus R2 plus R3. Now lets look at the condition where t is equals to infinity to get our other value, that is Vc of t at t is equals to infinity. We're going to draw our circuit one more time. And this time for the condition or the switch has been thrown and creates a short circuit Around our resistor R1. We still have resistors R2 and R3 in the circuit and we still have our capacitor as well. So we're going to draw those in. But we know that at t is equal to infinity. We're at another steady state condition and our capacitor is going to act like an open circuit as it does in all DC circuits that are in a steady state. This is Vc at t is equal to infinity. You have R1, R2, and R3 and V sub s in our circuit. So now, if we look at this circuit, we can also easily see what DC at time t is equal to infinity is. It's again going to be the voltage source dropped across. In this case, R2 and R3 because R1 has been short in it. There's a short circuit and parallel with R1 which essentially takes all current flow away from R1 and it'll travel through the short circuit instead. So Vc at t is equal to infinity is going to be V sub s times R3 divided by R2 plus R3. So to find our two unknown quantities Vc at t is equals to 0 minus and t is equal to infinity, we first drew the circuits for those two conditions. And then we use the properties of the capacitor at a DC steady state circuit in order to draw our circuits and solve for those two conditions.