We need to use a graph to find a number δ such that if |x−π4|<δ then |tanx−1|<0.2
First, we will get the values of x that intersect at the given curve to their corresponding y values. Let xL and xR
are the values of x from the left and right of π4 respectively.
y=tanxLy=tanxR0.8=tanxL1.2=tanxRxL=tan−1[0.8]xR=tan−1[1.2]xL=0.6741xR=0.8761
Now, we can determine the value of δ by checking the values of x that would give a smaller distance to π4.
π4−xL=π4−0.6747=0.1107π4−xR=0.8761−π4=0.0907
Hence,
δ≤0.0907
This means that by keeping x within 0.0907 of π4, we are able to keep f(x) within 0.2 of 1.
Although we chose δ=0.0907, any smaller positive value of δ would also have work.
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