Hello
I want to learn all methods for solving limits
can someone help me?
One of the more ingenious methods of evaluating limits is using the Sandwich Theorem.
One of the most famous limits is evaluated this way: .
This requires using some knowledge of the unit circle.
The red length is , the green length is and the purple length is .
Notice that the area of the sector is a little larger than that of the smaller triangle, and a little smaller than that of the larger triangle. As you make , the three areas will end up becoming equal.
So
, and since
.
It should be clear after being sandwiched that .
[Of course, to be pedantic, this only evaluates the right hand limit (i.e. making from the first quadrant where ), but the left hand limit is evaluated in a near identical manner.]