What Are Limits Used For In Calculus What Are Limits Used For In Calculus Limit and continuity in Calculus are two of the difficult topics in Mathematics. You should be familiar with them when you are going to learn calculus. The meaning of limits in calculus is to show the function value and it is the essential part of the calculus. In the function of limits we also consider two different topics :laws of limits and left and right hand limit. The expression of limits of function is shown by following formula :lim x-->c f ( x ) = L OR f ( x ) → L as x → c where f ( x ) is a real value function and c is a real number. Let’s take an example to understand it :- the function a2 - 9 / a – 3 that is not defined at a = 3, for other value of a, it generalize as a2 - 9 / a – 3 = (a – 3) ( a + 3 )/ a - 3 = a + 3 and where a → 3 so by simplify it we get a2 - 9 / a – 3 → 6 as a --> 3
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The meaning of the term continuity is same as we use in our daily life. When we say that a function f ( x ) is a continuous at a point x = a it means that at point ( a, f ( a ) ) the graphs of the function has no holes or graphs. That is graph is unbroken at point ( a, f ( a ) ). The definition of continuity at a point is, a function f ( x ) is said to be continuous at a point x = a of its domain, if lim x--->a f ( x ) = f ( a ) thus ( f ( x ) is continuous at x = 4 ) lim x--->a f ( x ) = f ( a ) lim x--->a - f ( x ) = lim x--->a + f ( x ) = f ( a ) If f ( x ) is not continuous at a point x = a then it is said to be discontinuous at x = a. A function f ( x ) is said to be left continuous or continuous from the left at x = a, if 1. lim x--->a - f ( x ) exists and 2. lim x--->a - f ( x ) = f ( a ) A function f ( x ) is said to be right continuous or continuous from the left at x = a, if 1. lim x--->a + f ( x ) exists and 2. lim x--->a + f ( x ) = f ( a ) so f ( x ) is continuous at x = a if it is both left as well as right continuous at x = a.
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The definition of a limit is not concerned with value of f(x) when, x=c. So, we care about the values of f(x) when x is close to c, on either the left side or right side. Now, we move on to rules regarding limits. Limits have some rules which are useful when we solve different limit problems. First Rule: First rule is called as the constant rule. In this rule we state- if we have f(x) =b (where f is constant for all x) then, the limits as x approaches c must be equal to b. It will always depend on the value of if x2 will grow more rapidly with respect to x as x increases then the function will surely tend towards the positive infinity Now, let’s talk about the degree of the function, it can be defined as the highest power of variable for example: 5x2 +6x+7, In this x2 has highest power as two so degree of the function will be 2. The degree of the function can be negative or positive. If degree of the function, is greater than 0 then, limit will always be positive. If degree of the function is less than 0 then the limit will be 0.
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Published on May 14, 2012
laws of limits and left and right hand limit. The expression of limits of function is shown by following formula :- Limit and continuity in...