CompuChip said:Do [square] and {curly} brackets have some special meaning to you? Because
[tex]\lim_{x \to 0} \frac{a}{x} \frac{b}{x} \neq \frac{x}{a} \left( \frac{b}{x} - \frac{b}{x} \right)[/tex]
doesn't really make sense to me.
Kartik. said:Lim(x-->0) x/a[b/x] can be written as x/a(b/x-{b/x})
how can we write this as
lim(x-->0)
(b/x -b/a({b/x}/{b/x}))?
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a specific value, known as the limit point. It represents the value that the function is approaching, or tending towards, as the input gets closer and closer to the limit point.
The limit of a function can be calculated using various methods, including algebraic manipulation, graphical analysis, and the use of limit laws and theorems. In general, to calculate a limit, we evaluate the function at values approaching the limit point from both the left and right sides and check if the resulting values are approaching the same value. If they are, that value is the limit of the function at that point.
Limits and continuity are closely related concepts in calculus. A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. In other words, a function is continuous if it has no breaks, holes, or jumps in its graph. Conversely, if a function is not continuous at a point, it means that the limit of the function at that point does not exist.
There are three main types of discontinuities: removable, jump, and infinite. A removable discontinuity occurs when there is a hole in the graph of a function at a specific point, but the limit of the function at that point exists. A jump discontinuity occurs when there is a sudden jump or gap in the graph of a function at a specific point, and the limits from the left and right sides are not equal. An infinite discontinuity occurs when the limit of a function approaches positive or negative infinity at a specific point.
Limits and continuity play a crucial role in calculus because they allow us to describe the behavior of a function at a specific point, even if the function is not defined at that point. They also help us to determine the existence of derivatives, which are essential in many applications of calculus, such as optimization, rates of change, and curve sketching. Additionally, the concepts of limits and continuity are fundamental in understanding the concept of a derivative, which is the cornerstone of calculus.