Finding the limit of one function can be considered easy depending on the complexity of the function. But what happens when we want to combine two or more functions? Then some of the methods we used before can make finding the limit more complicated. In order to work with these types of limits in a more simplified way, we need to use different properties of limits. One of these properties is the property of taking the limity of the addition or substraction of two functions. We say that if lim f(x)=L as x→a   and   lim g(x)=M as x→a. Then  lim [f(x)-g(x)] =L-M as x→a

Claim:  lim [f(x)-g(x)] =L-M   as   x→a

Proof:

|f(x)-L|< ε/2 for  δ_1 and  |g(x)-M|<ε/2 for δ_2

We want to prove that lim [f(x)-g(x)] =L-M   as  x→a    i.e.   |f(x)-L|-|g(x)-M|<ε   for the smallest  δ between δ_1 and δ_2

|f(x)-L-[g(x)-M]|=|[f(x)-L]+[-g(x)+M]|

By triangle inequality |[f(x)-L]+[-g(x)+M]| ≤ |f(x)-L|+|-g(x)+M|

Note that |-g(x)+M|=|g(x)-M|

|f(x)-L- ( g(x)-M )|≤ |f(x)-L|+|g(x)-M|

We know that |f(x)-L|< ε/2  and  |g(x)-M|<ε/2

|f(x)-L- ( g(x)-M )|< ε/2 +ε/2

|f(x)-L- ( g(x)-M )|< ε  

Your roommate is a baking lover. She is baking a cake for your birthday. You are a curious person and decide to take a look at the cake while it is in the oven. When you take a look at your cake you realize three things. First, your roommate chose a rectangular prism cake tin. The base of the tin is a rectangle of size 20x30cm. The height of the tin is 10cm. However, the tin is only half full. Second, you see that due to the baking soda, the upper surface of the cake is going up at a rate of 0.5cm/min. Third, your roommate chose to prepare a chocolate cake. You want to know the rate at which the volume of the cake is changing when the tin is full.

The volume of a rectangular prism is given by:

V=Area of the base x height

Since the volume is changing only due to the change in height. the rate of change of the volume, by implicit differentiation, is given by:

dV/dt = (20cm) x (30cm) x dh/dt

dV/dt = 600cm x 0.5cm²/min

dV/dt = 300 cm³/min