use the method in example 23.2 to prove the question. Example 23.2. Let f(r) be any linear function: f(r) = mr+b wherem and b are fixed real numbers. (The domain of f is all of R. ) Let abe any real number. I claimlim f(x) = f(a).(As we will see, this is equivalent to the property that f is "continuous"at I = a.) Pick E > 0.(Scratch work: We need |f(r) – f(@)I < ɛ and this is equivalentto m(I – a)| < ɛ which in turn (provided m + 0) is equivalent to|r- al < . If m = 0, then |f(r) – f(a)| < ɛ is automatic.)We proceed in cases.590. Then set 8Case I: Suppose m =you like). If 0 < |I- a| < 8, then f is defined at I and |f(x) – f(a)| =|6 – 6| = 0 < e. This proves lim,a f(1) = f(a).Case II: Suppose m + 0. Set 8 = : If 0 < |I – al < 8, then f is10100 (or any positive number%3Ddefined at I and|f(r)-f(a)| = |mr+b-(ma+b)| = |m(r-a)| = |m||r-a| < |m|8 = e.This proves lim,a f(1) = f(a). Let a > 0. Using just the e – ở definition of the limit of a function, prove thatlim Va = va.|I – a|VaTip: Use |VI - Va] =

Name: use the method in example 23.2 to prove the question. Example 23.2. Le
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: use the method in example 23.2 to prove the question. Example 23.2. Let f(r) be any linear function: f(r) = mr+b wherem and b are fixed real numbers. (The domain of f is all of R. ) Let abe any real number. I claimlim f(x) = f(a).(As we will see, thi