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The concept of limits in calculus arises from understanding function behavior as a parameter approaches a specific value []. For a continuous function, like $f(x) = x^2$, the limit as $x$ approaches a value $A$ is simply $f(A)$ []. However, limits become crucial when dealing with functions that have discontinuities, such as vertical asymptotes or holes, where direct evaluation at a point is impossible []. For instance, if a function has a removable discontinuity at $x=1$ due to a factor like $(x-1)$ in the denominator, the limit as $x$ approaches 1 can still be found by simplifying the function and evaluating the simplified form []. This is because limits describe the function's behavior *approaching* a point, not its value *at* that point [].
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Video summary will appear here after you start watching
The concept of limits in calculus arises from understanding function behavior as a parameter approaches a specific value []. For a continuous function, like $f(x) = x^2$, the limit as $x$ approaches a value $A$ is simply $f(A)$ []. However, limits become crucial when dealing with functions that have discontinuities, such as vertical asymptotes or holes, where direct evaluation at a point is impossible []. For instance, if a function has a removable discontinuity at $x=1$ due to a factor like $(x-1)$ in the denominator, the limit as $x$ approaches 1 can still be found by simplifying the function and evaluating the simplified form []. This is because limits describe the function's behavior *approaching* a point, not its value *at* that point [].