The previous example demonstrated it can sometimes be

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Unformatted text preview: ∆ in the right or in the left x direction. When the tangent of the angle at which the tangent line at point x0 to When the graph of function y = f ( x ) is intersecting the horizontal axis depends on the sign of ∆ x , the derivative of f ( x ) at x0 is not defined. In that case we can talk about the “left-hand side slope” and the In “right-hand side slope” or, more formally, the left-side and right-side left-side derivatives. derivatives. Notation for the right-side derivatives: f ′( x0 + 0 ) = lim + f ( x0 + ∆x ) − f ( x0 ) ∆x f ′( x0 − 0 ) = lim− f ( x0 + ∆x ) − f ( x0 ) ∆x x → x0 The left-side derivates: x → x0 12 An Example x, x ≥ 0 Consider a function y = f ( x ) = x = − x , x < 0 This function has a kink at (0,0): Let us compute the right-side Let derivative at point x0 = 0 : f ′( 0 + 0 ) = lim+ x →0 (0,0) 0 + ∆x − 0 =1 ∆x Now compute the left-side Now derivative at point x0 = 0 : f ′( 0 − 0 ) = lim+ x →0 0 − ∆x − 0 = −1 ∆x Even if the derivative of x does not exist at x=0, the left-side Even and right-side derivatives do exist and are well defined. 13 Left­Sided and Right­Sided Limits Limits are not only taken for the difference quotients in order to compute Limits the derivatives. We can talk in general about a limit of any function of any variable: any lim g ( v ) = g ( v0 ) = q0 v→ 0 v Limits are answering questions of the type, “what value does variable q=g(v) approach as variable v approaches v0 ? q=g(v) In some cases the answer depends on the direction in which v In approaches v0 The left-side limit of q is symbolized by: g ( v0 − 0) = vlim g ( v ) The − → v0 The right-side limit of q: g ( v0 + 0 ) = lim g ( v ) + v →v0 14 Graphical Illustration of One­Sided Limits q=g(v) 2, v < 5 This is a step function: q = g ( v ) = 1, v ≥ 5 As v approaches 5 from the left, the left-side limit As of g(v) is 2. of However, as v approaches 5 from the right, However, the right-side limit of g(v) is 1. the 2 The limit proper of g(v) at v=5 does not The exist, but the two one-sided (unequal) limits do. limits 1 5 v 15 Evaluation of the Limits () To evaluate a limit of function f x at point x0 it suffices to make sure To that the right- and the left-side limits at this point are equal to each other. This would be the case e.g. for This y = 2 + x2 However, what do we do when However, x0 and any real x0 is not even in the domain of f ( x ) ? Consider the following function that is not defined at x=1: 1− x2 y = f ( x) = 1− x We can still evaluate the value of the limit at x=1 even if the We function is not defined at x=1 (since you can’t divide by zero) by simplifying the expression: simplifying 1 − x 2 (1 − x ) (1 + x ) y = f ( x) = = = 1 + x, x ≠ 1 1− x 1− x lim f ( x ) =2 x→ 1 16 Ratio of Two Infinities Consider finding the limit of f ( x ) = Consider It is difficult to evaluate f It two infinities is equal to. 2x + 5 at x0 = + ∞ x +1 ( + ∞ ) since one is not sure what the ratio of Once a...
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