Graphical Approach to Limits. Note that the left and right hand limits and f(1) = 3 are all different. Note that the left hand limit and f(1) = 2 are equal. In this example, the limit when x approaches 0 is equal to f(0) = 1. This graph shows that as x approaches - 2 from the left, f(x) gets smaller and smaller without bound and there is no limit. As x approaches - 2 from the right, f(x) gets larger and larger without bound and there is no limit.
http://www.analyzemath.com/calculus/limits/introduction.html
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below.
https://en.wikipedia.org/wiki/Limit_of_a_function
The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function.
https://en.wikipedia.org/wiki/Limit_of_a_function
1 The limit of 1/ x is zero as x approaches infinity; the limit of (x − 1) 2 is zero as x approaches 1.  a number such that the absolute value of the difference between terms of a given sequence and the number approaches zero as the index of the terms increases to infinity.
http://www.dictionary.com/browse/limit
Part of the Mathematics glossary: In mathematics, a limit is a value toward which an expression converges as one or more variables approach certain values. Limits are important in calculus and analysis. Consider the limit of the expression 2 x + 3 as x approaches 0. It is not difficult to see that this limit is 3, because we can assign the value 0 to the variable x and perform the calculation directly.
http://whatis.techtarget.com/definition/limit
In both cases as x approaches 1, f(x) approaches 4. Intuitively, we say that limx→1 f(x) = 4. NOTE: We are talking about the values that f(x) takes when x gets closer to 1 and not f(1). In fact we may talk about the limit of f(x) as x approaches a even when f(a) is undefined. Example 2: Let g(x) = sin x / x and compute g(x) as x takes values closer to 0. We consider values of x approaching 0 from the left (x < 0) and values of x approaching 0 from the right (x > 0). Here we say that limx→0 g(x) = 1.
http://www.analyzemath.com/calculus/limits/introduction.html
Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1. In other words, the limit of (sin x)/x as x approaches zero equals 1.
https://en.wikipedia.org/wiki/Limit_of_a_function
Limits form the basis for calculus, where a number L is defined to be the limit approached by a function f(x) as x approaches a if, for every positive number ε, there exists a number δ such that |f(x)-L| < ε if 0 < |x-a| < δ.
http://www.dictionary.com/browse/limit
The limit of a function f(x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f(x) remain within the target distance.
https://en.wikipedia.org/wiki/Limit_of_a_function
a number such that the value of a given function remains arbitrarily close to this number when the independent variable is sufficiently close to a specified point or is sufficiently large. The limit of 1/ x is zero as x approaches infinity; the limit of (x − 1) 2 is zero as x approaches 1.
http://www.dictionary.com/browse/limit
what is limit
A value toward which an expression converges as one or more variables approach certain values is called as a limit..   Limit is a value toward which an expression converges as one or more variables approach certain values.