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Click to edit Master title style 1 Click to edit Master title style BASIC CALCULUS Ms. Angel Grace R. Jaen 2 Click to edit Master title style Let us pray for the strength and protection of all our frontliners especially our medical frontliners. 3 3 Click to edit Master title style CHAPTER I WEEK 1 Limits and Continuity Evaluate the limits of the functions by using tabular and graphical methods. 4 4 Click to edit Master title style Letβs have an activity! 5 5 y = 2x+3 Click to edit Master title style What is the value of y, if x = 2? A. 7 B. 4 C. 2 6 6 y = 2x+3 Click to edit Master title style What is the value of y, if x = -2? A. - 7 B. - 3 C. - 1 7 7 πβπ π= π+π Click to edit Master title style What is the value of y, if x = -3? A. 0 B. undefined C. 6 8 8 πβπ π= π+π Click to edit Master title style What is the value of y, if x = 1? A. 0 B. undefined C. 6 9 9 π+π π= π+π Click to edit Master title style What is the value of y, if x = 2? A. 2 B. 1 2 C. 3 10 10 Click edit Master title style ThetoTabular Method Consider the linear function: f(x) = 2x β 3 Let us say that we want to determine the limit of f(x) as the values of x approach to 1. We are interested in looking at the behavior of the f as the values of x get closer and closer to the number 1. 11 11 Click to edit Master title style Table 1 shows the value of x approaching the number of 1 from the left; that is, the value of x are getting closer and closer to 1, but they are all less than 1. How do the values of y = f(x) behave as the result? They also get closer to the number -1. x 0 0.5 0.75 0.8 0.9 0.99 0.999 0.9999 F(x) = 2x - 3 -3 -2 -1.5 -1.4 -1.2 -1.02 -1.002 -1.0002 12 12 Click to edit Master title style In Table 2, we make the values of x approach 1 from the right. This time, the values of x are getting closer and closer to 1, but they are all greater than 1. Then, we see that the resulting values of f(x) also approach the number -1. x 2 1.5 1.25 1.2 1.1 1.01 1.001 1.0001 F(x) = 2x - 3 1 0 -0.5 -0.6 -0.8 -0.98 -0.998 -0.9998 13 13 Click to What have edityou Master observed? title style x 0 0.5 0.75 0.8 0.9 0.99 0.999 0.9999 F(x) = 2x - 3 -3 -2 -1.5 -1.4 -1.2 -1.02 -1.002 -1.0002 x 2 1.5 1.25 1.2 1.1 1.01 1.001 1.0001 F(x) = 2x - 3 1 0 -0.5 -0.6 -0.8 -0.98 -0.998 -0.9998 14 14 Click to edit Master title style Limits β’ Limits are a tool for reasoning about function behavior, and tables are a tool for reasoning about limits. One nice thing about tables is that we can get more precise estimates of limits than we'd get by eyeballing graphs. β’ When using a table to approximate limits, it's important to create it in a way that simulates the feeling of getting "infinitely close" to some desired x-value. 15 15 Click to edit Master title style Limits β’ Imagine we're asked to approximate this limit: π₯β2 lim 2 π₯β2 π₯ β 4 β’ Note: The function is actually undefined at x=2, because the denominator evaluates to zero, but the limit as x approaches 2 still exists. β’ Step 1: We'd like to pick a value that's a little bit less than x=2 (that is, a value that's "to the left" of 2 when thinking about the standard x-axis), so maybe start with something like x=1.9. x f(x) 1.9 0.2564 2 undefined 16 16 Click to edit Master title style β’ Step 2: Try a couple more x-values to simulate the feeling of getting infinitely close to x=2 from the left. x f(x) 1.9 0.2564 1.99 0.2506 1.999 0.25001 2 undefined 17 17 Click to Master title β’ Step 3:edit Approach x=2 style from the right just like we did from the left. We want to do this in a way that simulates the feeling of getting infinitely close to x=2. x f(x) β’ 1.9 0.2564 1.99 1.999 2.0001 0.2506 0.25001 0.24999 Note: We've removed x=2x=2x, equals, 2 from the table to save space, and also because it isn't necessary for reasoning about the limit value. 2.01 0.2494 2.1 0.2439 18 18 Click to edit Master title style Letβs have an activity! 19 19 Click to edit Master title style x 1 1.25 1.5 1.75 1.9 1.99 1.999 1.9999 f(x) = 4 + 2x 6 6.5 7 7.5 7.8 7.98 7.998 7.9998 x 3 2.75 2.5 2.1 2.01 2.001 2.0001 2.00001 lim π + ππ = π πβπ f(x) = 4 + 2x 10 9.5 9 8.2 8.02 8.002 8.0002 8.00002 20 20 ππ β π Click toxedit Master title style π(π) = 0 0.5 0.75 0.8 0.9 0.99 0.999 0.9999 x πβπ 0.5 0.5 0.35 0.3 0.17 0.0197 0.001997 0.00019997 lim ππ β π πβπ π β π 1.8 1.75 1.5 1.1 1.01 1.001 1.0001 1.00001 =π ππ β π π(π) = πβπ -11.2 -8.25 -2.5 -0.23 -0.02 -0.002 -0.0002 -0.00002 21 21 Click to edit Master title style Any question? 22 22 Click to edit Master title style β Life has no limitations, except the ones you make.β - Les Brown 23 23 Click to edit Master title style Thank You! 24