Example Problem 3. Finding Limits Analytically

p {margin: 0; padding: 0;} .ft00{font-size:20px;font-family:Arial;color:#000000;} .ft01{font-size:16px;font-family:ArialMT;color:#000000;} .ft02{font-size:16px;line-height:21px;font-family:ArialMT;color:#000000;} Finding Limits Analytically. Example Problem 3 We can see a piecewise function at work in the following example.So, x to the second power plus 4 and we'll do this for values greater than 2. And perhapswe'll do 3x plus 1 for values less than or equal to 2. We're interested in finding the limit as xapproaches 2 for this function, f of x.Looking at the graph, you notice that the function f(x) = x2 + 4 takes on the values 2, 4, and6 when x = 2, 3, and 4. This parabola on a coordinate plane and shade in the area under itfrom x = 2 to x = 5.The graph of the function f(x) = 2x + 8 is an open circle centered at (2,8). The graph of thefunction g(x) = 3x + 1 is an open right triangle with vertices at (1,0),(3,1),and an upper rightvertex of 3. The two sides do not meet. The one side, let's see here, this side should be anopen circle. So at this point, the limit does not exist. Limit does not exist because theleft-hand limit and the right-hand limit are not the same. Basically, what we saw is the y values added to these, which were found by evaluating bothpieces of the piecewise 4x equals 2, they come to different y values. So essentially, what wecalculated was the limit as x approaches 2 from the left. And this piece right here, becauseit's x is less than 2, that means x is left of 2, like 1.9, 1.99. That would fall right here. We will use the 3x + 1 function. This evaluates to be 7. We can also use the limit as xapproaches 2 from the right of this component, which is x squared plus 4, since we knowthat values greater than 2.1, 2.2, etc. would be bigger than 2. When you plug in 2 for x, you get 8. So as you can see, the limit as x approaches 2 from theleft of f(x) does not equal the limit as x approaches 2 from the right of f(x). Therefore, thelimit as x approaches 2 of f(x) does not exist. If these two values were the same, if thefunction's two sides did come together, then the limit would be whatever that value was.