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MM

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Problem 82

Proof Consider the line $f(x)=m x+b,$ where $m \neq 0 .$ Use the $\varepsilon$ -\delta definition of limit to prove that $$\lim _{x \rightarrow c} f(x)=m c+b$$

Answer

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## Discussion

## Video Transcript

and this problem, we want to show that the limit has X approaches. C of the function f of X equals MX plus B, where M is non zero is equal to M C plus B and we want to do this using that sound out a definition of the limit. So that means we have to show that for each up Floren greater than zero. There is a Delta created in Sarah such that this absolute value, the absolutely of after lex minus and secrets be is less than epsilon whenever if the absolute value of excellent sea is in between zero and daughter. So we want to find the appropriate daughter that will make this work. So first, let's no that the absolute value of f of X minus m c plus B when we were kind of right that out by replacing a vex with its definition is equal to a Mex plus B minus M c plus B quantity. Now only simplify this out. This gives us annex minus emcee since you'll distribute this negative sign to M C and B, and the beans will cancel out nicely there. So then you noticed that both of these terms here has an M factor so we can write this out, factoring out the end. So this will be the absolute value of M times X minus c and then one more step. Buy properties of absolute values. We're multiplying two things inside of an absolutely sign we can kind of split them up into a product about so values. So that will be equal to the absolute I them terms the absolute value of Excellency. So now we have a relationship between this quantity here and this quantity here. So what? We can do this. Let Delta equal Absalon divided by the absolute value them this way, when Delta takes this value, our backs Linus M C plus B is less than absolute, and then we're done.

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