![]() This function is undefined at x = 3, because the denominator goes to zero. When the behavior differs from right and left, the limit does not exist.ġ4 Evaluating Limits Graphically Limits that do not existį(x) increases or decreases without bound as x approaches c. (0, -1) When x < 0, Approaching from the left, This is called the left-hand limit. means that x lies in the interval (c – δ, c + δ) i.e., 0 0, there exists a δ > 0 such that if 0 0, -2.5 2.5 2 -2 Approaching zero from the right, (0, 1) This is called the right-hand limit. The ε-δ definition of a limit: ε (epsilon) and δ (delta) are small positive numbers. IMPORTANT POINT: -5 5 4 -4 even though f(1) is undefined f(c) is undefined -5 5 4 -4 even though f(c) ≠ L ![]() The existence or nonexistence of f(x) at c has no bearing on the existence of In fact, even if, the limit can still exist. If f(x) becomes arbitrarily close to a single number, L, as x approaches c from either side, the limit of f(x) as x approaches c is L. SO… If f(x) becomes arbitrarily close to a single number, L, as x approaches c from either side, then the limit of f(x) as x approaches c is L. The limit itself is a value of the function, i.e., its y value. The limit is taken as the input, x, approaches a specific value from either side. See other examples on pages of your textbook. As the number of rectangles increases (and their width decreases) the approximation improves. The limit process: Sum up the areas of multiple rectan-gular regions. Finding the area under a curve is a calculus problem. The Area Problem Finding the area of a rectangle is a precalculus problem. As the distance between the two points decreases, the slope gets more accurate. ![]() Q The limit process: Compute the slope of a line through P and another point, Q, on the curve. Finding the slope of a curve at a point, P, is a calculus problem. The Tangent Line Problem Finding the slope of a straight line is a precalculus problem. To model the velocity of an accelerating object, you need calculus.Ĥ TWO CLASSIC CALCULUS PROBLEMS that illustrate how limits are used in calculus ![]() Graphically Numerically Analytically What is Continuity? Infinite Limits This presentation Bring your graphing calculator to class!ģ WHAT IS CALCULUS? Calculus is the mathematics of changeĭynamic, whereas earlier mathematics is static a limit machine Precalculus Mathematics Limit Process Calculus As an example, an object traveling at a constant velocity can be modeled with precalculus mathematics. 2 LIMITS What is Calculus? What are Limits? Evaluating Limits ![]()
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