During many Calculus courses the Concept of Limits are taught to students by showing them how, but not why. The purpose of this essay is to discuss and teach the concepts of Limits and why they exist. Limits are very important in all fields involving any form of math, whether solving the acceleration of a rocket ship to analyzing and interpolation stock market data.
A Limit is the height a function Intends to reach at a given x value, whether or not it actually reaches it (Kelly 57). By using f(x), a, and L we say that, "the Limit of f(x), as x approaches a, equals L" (Stewart 91). .
Lim f(x) = L.
Which in English means, we can make f(x) as close to L as we can by taking x sufficiently close to a, but not equal to a (Stewart 91).
Now Limits have certain rules to follow if they want to be Limits. First, the Limit of a point only exists when the limits from the approaching the point from the left(-) equals the limit of the point from the right(+). This concept is called Left and Right Hand Limits.
Lim f(x) = L = Lim f(x) = L then Lim f(x) = L.
So now that we know when a Limit exists, we need to know when a limit doesn"t exist. Since, we already know about the rule of left and right hand limits so we"re one third of the way there. Another reason for a limit wouldn"t exist would be for a graph to break, if the pieces don"t meet up at an intended height. This causes the left and right hand limits to not be equal therefore causing the limit to not exist. Another reason would occur if a function increases or decreases indefinitely at a given x-value (Stewart 61). This occurs when a function has a vertical asymptote at x=a causing the function to increase or decrease indefinitely. Finally, the last reason for a limit to not exist would be for the function to wiggle from one point to the next, never reaching a single numeric value. This occurs with trigonometric functions.
There are many ways to solve for limits.