a. false $$ v $$ If $$ \lim _{x \rightarrow 1^{-}} f(x)=5 $$, then $$ \lim _{x \rightarrow 1} f(x)=5 $$. b. false $$ \checkmark \lim _{x \rightarrow 1^{-}} f(x)=5 $$, then $$ \lim _{x \rightarrow 1^{+}} f(x)=5 $$. c. true If $$ \lim _{x \rightarrow 1} f(x)=5 $$, then $$ \lim _{x \rightarrow 1^{-}} f(x)=5 $$. d. true $$ \lim _{x \rightarrow 1} f(x)=5 $$, then $$ \lim _{x \rightarrow 1^{+}} f(x)=5 $$. e. Select all true statements. Assume that all the limits are all taken at the same point. A. If the two-sided limit exists, then the left- and right-hand limits both exist and are equal. B. If the left- and right-hand limits both exist and are equal, then the two-sided limit exists. C. If the left-hand limit exists, then the two-sided limit exists. D. the right-hand limit exists, then the two-sided limit exists.

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**a. Analysis**

We are given that  
$$
\lim_{x \to 1^-} f(x) = 5
$$  
and asked if we can conclude  
$$
\lim_{x \to 1} f(x) = 5.
$$  
For the two-sided limit to exist, both the left-hand and right-hand limits must exist and be equal. Since only the left-hand limit is specified, the right-hand limit might be different or might not exist. Therefore, the statement is **false**.

---

**b. Analysis**

We are given that  
$$
\lim_{x \to 1^-} f(x) = 5,
$$  
and the statement claims this implies  
$$
\lim_{x \to 1^+} f(x) = 5.
$$  
No relation is given between the left-hand and right-hand limits of $$f(x)$$. Knowing the left-hand limit alone does not determine the right-hand limit. Thus, the statement is **false**.

---

**c. Analysis**

We are given that  
$$
\lim_{x \to 1} f(x) = 5.
$$  
By definition, if the two-sided limit exists and equals 5, both one-sided limits exist and are equal to 5. Hence,  
$$
\lim_{x \to 1^-} f(x) = 5.
$$  
This statement is **true**.

---

**d. Analysis**

Similarly, from  
$$
\lim_{x \to 1} f(x) = 5,
$$  
we know that  
$$
\lim_{x \to 1^+} f(x) = 5.
$$  
Thus, the statement is **true**.

---

**e. Analysis of the Statements**

**A.** "If the two-sided limit exists, then the left- and right-hand limits both exist and are equal."  
- This is **true** because the existence of the two-sided limit implies both one-sided limits exist and are equal.

**B.** "If the left- and right-hand limits both exist and are equal, then the two-sided limit exists."  
- This is **true** because if both one-sided limits exist and are equal, the two-sided limit exists and equals that common value.

**C.** "If the left-hand limit exists, then the two-sided limit exists."  
- This is **false** because the existence of only the left-hand limit is not enough; the right-hand limit might not exist or might differ.

**D.** "If the right-hand limit exists, then the two-sided limit exists."  
- This is **false** for the same reason as in C; the existing right-hand limit alone does not guarantee the existence of the two-sided limit.

**Conclusion for e:** Only statements **A** and **B** are true.
**a.** False **b.** False **c.** True **d.** True **e.** A and B are true.

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