Find the value of 5lim𝑥→13+⌊𝑥⌋−3lim𝑥→3−⌊𝑥⌋5 x→13 + lim ⌊x⌋−3 x→3 − lim ⌊x⌋, where ⌊𝑥⌋⌊x⌋ denotes the greatest integer less than or equal to 𝑥x.
Question
Find the value of 5limx→13+⌊x⌋−3limx→3−⌊x⌋5 x→13 + lim ⌊x⌋−3 x→3 − lim ⌊x⌋, where ⌊𝑥⌋⌊x⌋ denotes the greatest integer less than or equal to 𝑥x.
Solution
The given expression is 5lim(x→13) [⌊x⌋] - 3lim(x→3) [⌊x⌋].
The greatest integer function, also known as the floor function, gives the largest integer less than or equal to x.
Step 1: Evaluate the limits separately.
For the first limit, as x approaches 13, the greatest integer less than or equal to x is 13. So, lim(x→13) [⌊x⌋] = 13.
For the second limit, as x approaches 3, the greatest integer less than or equal to x is 3. So, lim(x→3) [⌊x⌋] = 3.
Step 2: Substitute the values of the limits into the expression.
= 513 - 33 = 65 - 9 = 56
So, the value of the given expression is 56.
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