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## Finding Limits Analytically. Example Problem 3

Find: lim_(x->2) f(x) [Graph]

Let f(x)={x^2+4; 3x+1; x>2, x<=2

lim_(x->2) f(x)

lim_(x->2) -3x+1=7 -> lim_(x->2)-f(x)/=lim_(x->2)+f(x)

lim_(x->2)+x^2 +4=8 -> :.

lim_(x->2) f(x) DNE

We can see a piecewise function at work in the following example.

So, x to the second power plus and we'll do this for values greater than 2.

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And perhaps

we'll do 3x plus 1 for values less than or equal to 2. We're interested in finding the limit as x

approaches 2 for 2 this function. f of x.

Looking at the at graph, you notice that the function f(x) = x2 + 4 takes on the values 2, 4, and

6 when x = 2, 3, and 4. This parabola on a coordinate plane and shade in the area under it

from 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 the

function g(x) = 3x + 1 is an open right triangle with vertices at (1,0), (3,1), and an upper right

vertex of 3. The two sides do not meet. The one side. let's see here. this side should be an

open circle. So at this point, the limit does not exist. Limit does not exist because the

left-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 both

pieces of the piecewise 4x equals 2, they come to different y values.

So essentially, what we

calculated was the limit as x approaches 2 from the left. And this piece right here, because

it'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 x

approaches 2 from the right of this component, which is x squared plus 4, since we know

that 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 : from the

left of f(x) does not equal the limit as x approaches 2 from the right of f(x). Therefore, the

limit as x approaches 2 of f(x) does not exist. If these two values were the same, if the

function's two sides did come together, then the limit would be whatever that value was.

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