How do you find the limit of #| x - 5 |# as x approaches #5#?

Answer 1

0

#abs(x-5)# is a continuous funtion so #lim_(x to 5) abs ( x-5) = abs (5 - 5) = 0#
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Answer 2

If you are doing this to prove that the function is continuous, rewrite using the definition of absolute value.

#absu = {(u," if ",u >= 0),(-u," if ",u < 0) :}#
With #u = x-5#, the condition #u >= 0# becomes #x-5 >= 0# which is equivalent to #x >= 5#. Similarly #u < 0#, becomes #x < 5#.

So,

#abs(x-5) = {(x-5," if ",x >= 5),(-(x-5)," if ",x < 5) :}#
#lim_(xrarr5^+)abs(x-5) = lim_(xrarr5^+)(x-5) = 0# and #lim_(xrarr5^-)abs(x-5) = lim_(xrarr5^-)(-(x-5)) = -(0)=0#
Since both one-sided limits are #0#, the limit is #0#.
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Answer 3

To find the limit of | x - 5 | as x approaches 5, we can evaluate the expression for values of x that approach 5 from both the left and right sides.

When x approaches 5 from the left side (x < 5), the expression simplifies to -(x - 5), which is equal to 5 - x.

When x approaches 5 from the right side (x > 5), the expression simplifies to x - 5.

Since the limit from both sides is the same, the limit of | x - 5 | as x approaches 5 is equal to 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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