Amelia Rose Smith 231233678 CARNEGIE MELLON UNIVERSITY Concepts of Limit and Continuity : Limit Characteristics  The average limit feature and the highest limit feature Two crucial limits in calculus properties are the maximum limit property and the average limit property. According to the average limit property, the limit of f(x) as x gets closer to a exists and equals L if f(x) near L as x near an on both the left or the right sides. This proves that the limit occurs and is identical to the common value since the equation gets a single quantity from either side of the limit point. According to the greatest limit property, if the limit of g(x) as x gets closer to an is L and that limit of f(x) as x approaching an also exists and is smaller or equivalent to L for values of x near a, then f(x) will always be lower than or equivalent to g(x) for values of x close a (except potentially at an itself). This implies that the upper bound of the lesser function is likewise smaller or equivalent to the limitation of the bigger functional at that point if we have both functions that are continuously less than or equivalent to one another at a limit point. The compression theorem or sandwich theorem are other names for the average limit condition. When a function fluctuates or displays a "hole" at its maximum point, this characteristic might be helpful. As x gets closer to 0, for instance, think about the equation: f(x) = sin(x)/x. We may use the mean limit property to demonstrate that the limit of f(x) as x gets nearer 0 is 1, despite the fact that we are aware that sin(x)/x oscillates among -1 and 1 as x approaches 0. This may be achieved by first realizing that for any x≠ 0, -1/x≤ sin(x)/x≤ 1/x, and then finding the limit of the bottom and upper limits as x approaches 0. This results in the conclusion that f(x) has a limit of 1 as x gets closer to 0. When a function fluctuates or has a "hole" at its lowest point, this characteristic might be helpful. As x gets closer to 0, for instance, think about the equation: f(x) = sin(x)/x. We may use the average limit property to demonstrate that the limit of f(x) as x approaches 0 is 1, despite the fact that we are aware that sin(x)/x oscillates between -1 and 1 as x approaches 0. We may accomplish this by realizing that for any x≠ 0, -1/x≤ sin(x)/x ≤1/x, and then taking . When comparing the boundaries of two operations that aren't equal but always fall inside a certain range, the highest limit attribute is helpful. As x gets closer to 0, for instance, think about the formulas: f(x) = x^2 and g(x) = x^3. We can utilize the greatest limit property to demonstrate that the maximum value of f(x) as x gets nearer 0 is also 0. We already know that g(x) increases more quickly than f(x) as x reaches 0. This is because the maximum of g(x) as x becomes 0 and f(x) g(x) for any x approaching 0 (except from at 0 itself).Consequently, when x gets closer to 0, the upper bound of f(x) is also 0. In calculus, these features are useful resources for locating and assessing limits. In situations when substitution directly or algebraic manipulation are impractical, they enable us to evaluate limits.the intersection of the lower and higher boundaries as x gets closer to 0. This results in the conclusion that f(x) has a limit of 1 as x approaches 0.  Infinite limit feature In calculus, the concept of bounds at infinite is a crucial idea. It claims that the limit of a function f(x) as x gets closer to infinite or zero does not exist if f(x) randomly grows or shrinks as x approaches infinite or negative infinity. Formally speaking, if there is a N > 0 which means if x > N, then f(x) > M exists for any M > 0, then we say that the maximum value of f(x) as x gets nearer infinity is infinity. As a result, the values of f(x) can take on any size as x increases in size. In a similar vein, if there exists a N< 0 such that if x< N, then f(x)< M for all M< 0, then can say that the maximum of f(x) as x near minus infinite is negative infinity. As a result, the coefficients of f(x) grow indefinitely tiny as x becomes progressively less positive (i.e., negative and huge in magnitude). The maximum value of f(x) as x gets closer to infinity or zero infinity occurs and is finite, however, if the value of f(x) fails to grow arbitrarily large or tiny as x near infinite or negative infinity. When confronted with functions that increase or decrease rapidly, such as f(x) = e^(x) or g(x) = 1/x, the idea of limitations at infinity is very helpful. In these situations, we may assess limits of function that can't be evaluated by substitution directly and identify the actions of a function as x nears infinity or infinity negative by using the characteristics of limits at infinity. It is also possible to analyze the final behavior of functions using limits at infinity. We may determine if an equation has an asymptote that is horizontal and, if so, its value, by looking at the limit of the function as x gets closer to infinity or negative infinity. Take the equation f(x) = (2x^2 + 3x - 1)/(x^2 - 4x + 3) as an illustration. Both the numerator as well as the denominator of the equation increase without bound as x gets closer to infinity. The function, however, develops more quickly than x^2 because the angle of the fraction is higher than the amount of the denominator. when a result, when x gets closer to infinity, the maximum value of f(x) is infinite. On the other hand, both the denominator and numerator of the formula becomes extremely huge negative values when x approaches negative infinity. The function gets increasingly positive as x gets closer to zero because the degree in the denominator is still bigger than the level of the remainder. when a result, when x gets closer to negative infinity, the upper bound of f(x) is also infinity. This indicates that there is no horizontal asymptote for the function. The idea of limits at infinity may also be applied to multivariable functions, it is important to note. Here, rather than only along one axis, we study limitations as the variables go closer and closer to infinity. Because of this, the idea of the limit at infinity is unbounded and the function increases unrestricted in all directions. The investigation of surface and volumes, as well as optimization issues involving functions of many variables, benefit from the multivariable calculus idea of limits at infinity.  Limit rules for subtraction, addition and multiplication We may work with limits algebraically thanks to the limit arithmetic rules, which are a collection of characteristics. The multiplication rule, subtraction rule, and addition rule are some of these rules. According to the addition rule, if the boundaries of two functions, f(x) and g(x), are present as x gets near a, then f(x) + g(x) has a limit that is present and is equivalent to the sum of the limits of the two individual functions. In other words, lim[x-->a] (f(x) + g(x)) = L + M if lim[x-->a] f(x) = L and lim[x-->a] g(x) = M. We may estimate the boundaries of function that are summation of smaller functions using this technique. The distinction between the subtraction rule and the addition rule is that the former applies to differences rather than sums.According to this, if the limits of two functions, f(x) and g(x), exist as x gets closer to a, then f(x) - g(x) has a limit that is equal to the difference between their limits. In other words, lim[x-->a] (f(x) - g(x)) = L - M if lim[x-->a] f(x) = L and lim[x-->a] g(x) = M. We may assess the limits of functions that differ from simpler functions using this rule. According to the multiplication rule, if the limits for both functions, f(x) and g(x), exist as x gets closer to a, then the limit of their combined value, f(x)g(x), must likewise exist and be equal to the sum of their limits. To put it another way, lim[x-->a] (f(x)g(x)) = LM if lim[x-->a] f(x) = L and lim[x-->a] g(x) = M. We may assess the limitations of operations that are the result of smaller functions using this approach. It's crucial to remember that these guidelines only take effect when each function's bounds are met. To assess limits algebraically in some situations, we might need to utilize extra methods like factoring or rationalizing.