Investigating Limit of Piecewise Function

nycmathguy
Homework Statement
Graphs & Functions
Relevant Equations
Piecewise Functions
Use the graph to investigate

(a) lim of f(x) as x→2 from the left side.

(b) lim of f(x) as x→2 from the right side.

(c) lim of f(x) as x→2.

Question 20

For part (a), as I travel along on the x-axis coming from the left, the graph reaches a height of 4. The limit is 4. It does not matter if there is a hole at (2, 4), right?

For part (b), as I travel along on the x-axis coming from the right, the graph reaches a height of 2. The limit is 2. It does not matter if there is a hole at (2, 2), right?

For part (c), LHL DOES NOT EQUAL RHL.

I conclude the limit does not exist.

You say?
 

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Mark44 said:
Yes to all.

I got another question right. It's a miracle. I need to understand this limit idea better. I know that limits in calculus 3 are more involved.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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