question 11 define the sequence of funetions ix if xl fz 2 2v1 if cl 4 and f x il sketch a graph

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Question 1.1. Define the sequence of functions If |x| < 4, then f(x) = âˆš(x^2 + 2âˆš1) and f(x) = âˆš(x^2 + 2âˆš1) if |x| â‰¥ 4. Sketch a graph that includes both f1 and f in the same picture. For example, one could just graph maybe f2 and f3 and it would also be cool, but not mandatory, to graph all of f1, f2, f3, and f4 as well as f to try to see the pattern, at least visually. Prove that f is not differentiable at x = 0 (You will most likely have to use the definition of the derivative for this one. Notice that you do not have a result at your disposal that says anything about right and left limits of f'. That means you need to invoke the definition directly.) Prove that for all n âˆˆ N, fn is a continuous function on R. You may invoke all of the theorems in sections 17 and 18 for the case that x â‰ +4. But you need to use the definition of continuity for the case x = 1. You are allowed to use the fact that âˆ€Îµ > 0, |x| < Îµ â‡’ |fn(x)| < 2^(2n). The reason this is relevant is that because for a generic sequence (Sk) with limk Sk, you don't have any information on which terms in (Sk) satisfy, for example, Sk < 4 and which satisfy Sk > 4, and thus it becomes complicated to evaluate f(Sk). Instead, try the Squeeze theorem.

          Question 1.1. Define the sequence of functions
If |x| < 4, then f(x) = âˆš(x^2 + 2âˆš1)
and
f(x) = âˆš(x^2 + 2âˆš1) if |x| â‰¥ 4.
Sketch a graph that includes both f1 and f in the same picture. For example, one could just graph maybe f2 and f3 and it would also be cool, but not mandatory, to graph all of f1, f2, f3, and f4 as well as f to try to see the pattern, at least visually. Prove that f is not differentiable at x = 0 (You will most likely have to use the definition of the derivative for this one. Notice that you do not have a result at your disposal that says anything about right and left limits of f'. That means you need to invoke the definition directly.)
Prove that for all n âˆˆ N, fn is a continuous function on R. You may invoke all of the theorems in sections 17 and 18 for the case that x â‰  +4. But you need to use the definition of continuity for the case x = 1. You are allowed to use the fact that
âˆ€Îµ > 0, |x| < Îµ â‡’ |fn(x)| < 2^(2n).
The reason this is relevant is that because for a generic sequence (Sk) with limk Sk, you don't have any information on which terms in (Sk) satisfy, for example, Sk < 4 and which satisfy Sk > 4, and thus it becomes complicated to evaluate f(Sk). Instead, try the Squeeze theorem.

question 11 define the sequence of funetions ix if xl fz 2 2v1 if cl 4 and f x il sketch a graph that includes both f and f in the same pictue for example jon could just graph maybe fz and f 07784

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