![]() To determine continuity at a point, we use the formal definition that a function is continuous at a point c if and only if the limit of the function as x approaches c exists, the function is defined at the point c, and the limit of the function as x approaches c is equal to the function's value at c. In conclusion, continuity at a point is an essential concept in Calculus that builds upon our understanding of limits and discontinuities. Therefore, the function f(x) = x^2*e^x is continuous at x = 0. The limit of f(x) as x approaches 0 is equal to the function's value at x = 0, which is 0.The limit of f(x) as x approaches 0 exists and is equal to 0.Therefore, the function f(x) = sin(1/x) is not continuous at x = 0.Įxample 7: Is the function f(x) = x^2*e^x continuous at x = 0? The limit of f(x) as x approaches 0 is not equal to the function's value at x = 0, which is 0.The limit of f(x) as x approaches 0 does not exist, as the function oscillates between positive and negative values.Therefore, the function f(x) = x^3/(x-1) is not continuous at x = 1.Įxample 6: Is the function f(x) = sin(1/x) continuous at x = 0? The limit of f(x) as x approaches 1 is not equal to the function's value at x = 1 which is undefined. ![]()
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