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