Question

a) Let \( \epsilon=1 / 2 \). Show that no possible \( \delta>0 \) satisfies the following condition: For all \( x, 0<|x-1|<\delta \quad \Longrightarrow \quad|f(x)-2|<1 / 2 \). That is, for each \( \delta>0 \) show that there is a value of \( x \) such that \[ 0<|x-1|<\delta \quad \text { and } \quad|f(x)-2| \geq 1 / 2 . \] This will show that \( \lim _{x \rightarrow 1} f(x) \neq 2 \). b) Show that \( \lim _{x \rightarrow 1} f(x) \neq 1 \). c) Show that \( \lim _{x \rightarrow 1} f(x) \neq 1.5 \).

          a) Let \( \epsilon=1 / 2 \). Show that no possible \( \delta>0 \) satisfies the following condition:

For all \( x, 0<|x-1|<\delta \quad \Longrightarrow \quad|f(x)-2|<1 / 2 \).
That is, for each \( \delta>0 \) show that there is a value of \( x \) such that
\[
0<|x-1|<\delta \quad \text { and } \quad|f(x)-2| \geq 1 / 2 .
\]
This will show that \( \lim _{x \rightarrow 1} f(x) \neq 2 \).
b) Show that \( \lim _{x \rightarrow 1} f(x) \neq 1 \).
c) Show that \( \lim _{x \rightarrow 1} f(x) \neq 1.5 \).

a) Let ϵ=1 / 2. Show that no possible δ>0 satisfies the following condition:

For all x, 0<|x-1|<δ ⟹ |f(x)-2|<1 / 2.
That is, for each δ>0 show that there is a value of x such that

0<|x-1|<δ and |f(x)-2| ≥ 1 / 2 .

This will show that limx → 1 f(x) ≠ 2.
b) Show that limx → 1 f(x) ≠ 1.
c) Show that limx → 1 f(x) ≠ 1.5.

Added by Kristi A.

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