f230e2…: “Under the still river”

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Worked Out Examples Problem Videos

Remember that the previous practice problem tile had worked out example videos you can review to help with some of these problems if needed!

Practice Problems

1 : If you know that $\lim _{x \to \sage {p1target}} f(x) = \sage {p1c1}$ and $\lim _{x \to \sage {p1target}} g(x) = \sage {p1c2}$ , then evaluate the following limit:

$\lim _{x \to \sage {p1target}} \left ( \sage {p1c3} f(x) + \sage {p1c4}g(x) \right ) = \answer {\sage {p1ans}}$

Remember that you can use limit laws to rewrite $\lim _{x \to \sage {p1target}} \left ( \sage {p1c3} f(x) + \sage {p1c4}g(x) \right ) = \lim _{x \to \sage {p1target}}( \sage {p1c3}) \cdot \lim _{x \to \sage {p1target}}(f(x)) + \lim _{x \to \sage {p1target}}(\sage {p1c4}) \cdot \lim _{x \to \sage {p1target}}(g(x))$

2 : If you know that $\lim _{x \to \sage {p2target}} f(x) = \sage {p2c1}$ and $\lim _{x \to \sage {p2target}} g(x) = \sage {p2c2}$ , then evaluate the following limit:

$\lim _{x \to \sage {p2target}} \left ( \sage {p2c3} f(x) + \sage {p2c4}g(x) \right ) = \answer {\sage {p2ans}}$

Remember that you can use limit laws to rewrite $\lim _{x \to \sage {p2target}} \left ( \sage {p2c3} f(x) + \sage {p2c4}g(x) \right ) = \lim _{x \to \sage {p2target}}( \sage {p2c3}) \cdot \lim _{x \to \sage {p2target}}(f(x)) + \lim _{x \to \sage {p2target}}(\sage {p2c4}) \cdot \lim _{x \to \sage {p2target}}(g(x))$

3 : If you know that $\lim _{x \to \sage {p3target}} f(x) = \sage {p3c1}$ and $\lim _{x \to \sage {p3target}} g(x) = \sage {p3c2}$ , then evaluate the following limit:

$\lim _{x \to \sage {p3target}} \left ( \sage {p3c3} f(x) + \sage {p3c4}g(x) \right ) = \answer {\sage {p3ans}}$

Remember that you can use limit laws to rewrite $\lim _{x \to \sage {p3target}} \left ( \sage {p3c3} f(x) + \sage {p3c4}g(x) \right ) = \lim _{x \to \sage {p3target}}( \sage {p3c3}) \cdot \lim _{x \to \sage {p3target}}(f(x)) + \lim _{x \to \sage {p3target}}(\sage {p3c4}) \cdot \lim _{x \to \sage {p3target}}(g(x))$

For the next several problems, consider the following function $f(x) = \begin {cases} \sage {p4f1} & x<\sage {p4pivot1}\\ \sage {p4f2} &\sage {p4pivot1} \leq x < \sage {p4pivot2}\\ \sage {p4f3} &\sage {p4pivot2}<x\\ \end {cases}$ Determine if the following limits exist. If they do, compute their values.

4 : Does $\lim \limits _{x\rightarrow \sage {p4pivot1}^-} f(x)$ exist?

Yes. No.

4.1 : $\lim \limits _{x\rightarrow \sage {p4pivot1}^-} f(x) = \answer {\sage {p4anspvt1L}}$

5 : Does $\lim \limits _{x\rightarrow \sage {p4pivot1}^+} f(x)$ exist?

Yes. No.

5.1 : $\lim \limits _{x\rightarrow \sage {p4pivot1}^+} f(x) = \answer {\sage {p4anspvt1R}}$

6 : Does $\lim \limits _{x\rightarrow \sage {p4pivot2}^-} f(x)$ exist?

Yes. No.

6.1 : $\lim \limits _{x\rightarrow \sage {p4pivot2}^-} f(x) = \answer {\sage {p4anspvt2L}}$

7 : Does $\lim \limits _{x\rightarrow \sage {p4pivot2}^+} f(x)$ exist?

Yes. No.

7.1 : $\lim \limits _{x\rightarrow \sage {p4pivot2}^+} f(x) = \answer {\sage {p4anspvt2R}}$

8 : Compute $\lim \limits _{x\rightarrow \sage {p4pivot1}} f(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p4pivot1}} f(x) = \answer {\sage {p4anslim1}}$ .

9 : Compute $\lim \limits _{x\rightarrow \sage {p4pivot2}} f(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p4pivot2}} f(x) = \answer {\sage {p4anslim2}}$ .

For the next several problems, consider the following function $g(x) = \begin {cases} \sage {p5f1} & x<\sage {p5pivot1}\\ \sage {p5f2} &\sage {p5pivot1} \leq x < \sage {p5pivot2}\\ \sage {p5f3} &\sage {p5pivot2}<x\\ \end {cases}$ Determine if the following limits exist. If they do, compute their values.

10 : Does $\lim \limits _{x\rightarrow \sage {p5pivot1}^-} g(x)$ exist?

Yes. No.

10.1 : $\lim \limits _{x\rightarrow \sage {p5pivot1}^-} g(x) = \answer {\sage {p5anspvt1L}}$

11 : Does $\lim \limits _{x\rightarrow \sage {p5pivot1}^+} g(x)$ exist?

Yes. No.

11.1 : $\lim \limits _{x\rightarrow \sage {p5pivot1}^+} g(x) = \answer {\sage {p5anspvt1R}}$

12 : Does $\lim \limits _{x\rightarrow \sage {p5pivot2}^-} g(x)$ exist?

Yes. No.

12.1 : $\lim \limits _{x\rightarrow \sage {p5pivot2}^-} g(x) = \answer {\sage {p5anspvt2L}}$

13 : Does $\lim \limits _{x\rightarrow \sage {p5pivot2}^+} g(x)$ exist?

Yes. No.

13.1 : $\lim \limits _{x\rightarrow \sage {p5pivot2}^+} g(x) = \answer {\sage {p5anspvt2R}}$

14 : Compute $\lim \limits _{x\rightarrow \sage {p5pivot1}} g(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p5pivot1}} g(x) = \answer {\sage {p5anslim1}}$ .

15 : Compute $\lim \limits _{x\rightarrow \sage {p5pivot2}} g(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p5pivot2}} g(x) = \answer {\sage {p5anslim2}}$ .

For the next several problems, consider the following function $h(x) = \begin {cases} \sage {p6f1} & x<\sage {p6pivot1}\\ \sage {p6f2} &\sage {p6pivot1} \leq x < \sage {p6pivot2}\\ \sage {p6f3} &\sage {p6pivot2}<x\\ \end {cases}$ Determine if the following limits exist. If they do, compute their values.

16 : Does $\lim \limits _{x\rightarrow \sage {p6pivot1}^-} h(x)$ exist?

Yes. No.

16.1 : $\lim \limits _{x\rightarrow \sage {p6pivot1}^-} h(x) = \answer {\sage {p6anspvt1L}}$

17 : Does $\lim \limits _{x\rightarrow \sage {p6pivot1}^+} h(x)$ exist?

Yes. No.

17.1 : $\lim \limits _{x\rightarrow \sage {p6pivot1}^+} h(x) = \answer {\sage {p6anspvt1R}}$

18 : Does $\lim \limits _{x\rightarrow \sage {p6pivot2}^-} h(x)$ exist?

Yes. No.

18.1 : $\lim \limits _{x\rightarrow \sage {p6pivot2}^-} h(x) = \answer {\sage {p6anspvt2L}}$

19 : Does $\lim \limits _{x\rightarrow \sage {p6pivot2}^+} h(x)$ exist?

Yes. No.

19.1 : $\lim \limits _{x\rightarrow \sage {p6pivot2}^+} h(x) = \answer {\sage {p6anspvt2R}}$

20 : Compute $\lim \limits _{x\rightarrow \sage {p6pivot1}} h(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p6pivot1}} h(x) = \answer {\sage {p6anslim1}}$ .

21 : Compute $\lim \limits _{x\rightarrow \sage {p6pivot2}} h(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p6pivot2}} h(x) = \answer {\sage {p6anslim2}}$ .

For the next several problems, consider the following function $k(x) = \begin {cases} \sage {p7f1} & x<\sage {p7pivot1}\\ \sage {p7f2} &\sage {p7pivot1} \leq x < \sage {p7pivot2}\\ \sage {p7f3} &\sage {p7pivot2}<x\\ \end {cases}$ Determine if the following limits exist. If they do, compute their values.

22 : Does $\lim \limits _{x\rightarrow \sage {p7pivot1}^-} k(x)$ exist?

Yes. No.

22.1 : $\lim \limits _{x\rightarrow \sage {p7pivot1}^-} k(x) = \answer {\sage {p7anspvt1L}}$

23 : Does $\lim \limits _{x\rightarrow \sage {p7pivot1}^+} k(x)$ exist?

Yes. No.

23.1 : $\lim \limits _{x\rightarrow \sage {p7pivot1}^+} k(x) = \answer {\sage {p7anspvt1R}}$

24 : Does $\lim \limits _{x\rightarrow \sage {p7pivot2}^-} k(x)$ exist?

Yes. No.

24.1 : $\lim \limits _{x\rightarrow \sage {p7pivot2}^-} k(x) = \answer {\sage {p7anspvt2L}}$

25 : Does $\lim \limits _{x\rightarrow \sage {p7pivot2}^+} k(x)$ exist?

Yes. No.

25.1 : $\lim \limits _{x\rightarrow \sage {p7pivot2}^+} k(x) = \answer {\sage {p7anspvt2R}}$

26 : Compute $\lim \limits _{x\rightarrow \sage {p7pivot1}} k(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p7pivot1}} k(x) = \answer {\sage {p7anslim1}}$ .

27 : Compute $\lim \limits _{x\rightarrow \sage {p7pivot2}} k(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p7pivot2}} k(x) = \answer {\sage {p7anslim2}}$ .

For the next several problems, consider the following function $r(x) = \begin {cases} \sage {p8f1} & x<\sage {p8pivot1}\\ \sage {p8f2} &\sage {p8pivot1} \leq x < \sage {p8pivot2}\\ \sage {p8f3} &\sage {p8pivot2}<x\\ \end {cases}$ Determine if the following limits exist. If they do, compute their values.

28 : Does $\lim \limits _{x\rightarrow \sage {p8pivot1}^-} r(x)$ exist?

Yes. No.

28.1 : $\lim \limits _{x\rightarrow \sage {p8pivot1}^-} r(x) = \answer {\sage {p8anspvt1L}}$

29 : Does $\lim \limits _{x\rightarrow \sage {p8pivot1}^+} r(x)$ exist?

Yes. No.

29.1 : $\lim \limits _{x\rightarrow \sage {p8pivot1}^+} r(x) = \answer {\sage {p8anspvt1R}}$

30 : Does $\lim \limits _{x\rightarrow \sage {p8pivot2}^-} r(x)$ exist?

Yes. No.

30.1 : $\lim \limits _{x\rightarrow \sage {p8pivot2}^-} r(x) = \answer {\sage {p8anspvt2L}}$

31 : Does $\lim \limits _{x\rightarrow \sage {p8pivot2}^+} r(x)$ exist?

Yes. No.

31.1 : $\lim \limits _{x\rightarrow \sage {p8pivot2}^+} r(x) = \answer {\sage {p8anspvt2R}}$

32 : Compute $\lim \limits _{x\rightarrow \sage {p8pivot1}} r(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p8pivot1}} r(x) = \answer {\sage {p8anslim1}}$ .

33 : Compute $\lim \limits _{x\rightarrow \sage {p8pivot2}} r(x)$ . If it does not exist, write ’DNE’ $\lim \limits _{x\rightarrow \sage {p8pivot2}} r(x) = \answer {\sage {p8anslim2}}$ .