Chapter 2 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   Find ε in terms of δ for the function $f(x)=4x+1$

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2. Original problem

   Solve the limit $\underset{x\to \mathrm{\infty }}{lim}\frac{\frac{4x^3-3x}{x^3}}{\frac{5x^2+2x+5}{x^2}}$

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3. Original problem

   Find $\underset{x\to 4}{lim}f(x)=\{\begin{array}{rlr}\sqrt{x},& & ifx\ne 4\\ 4.& & ifx=4\end{array}$

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4. Original problem

   Solve $\underset{x\to 0}{lim}\frac{\cos (x)-1}{\sin (x)}$

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5. Original problem

   Simplify and solve $\underset{x\to 0}{lim}\frac{2(1-{\cos }^{2}x)}{2\sin x+\sin (2x)}$

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6. Original problem

   Solve the limit $\underset{x\to 0}{lim}\frac{\cos x\left(\cos x-1\right)+\sin x}{x^2}$

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7. Original problem

   Find $\underset{x\to 0}{lim}\frac{\sqrt{4-x}-2}{\sqrt{x+1}-1}$

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8. Original problem

   Find $\underset{x\to 1}{lim}\frac{\sqrt{x^2+3}-2}{\sqrt{x^2+8}-3}$

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9. Original problem

   What is $\underset{x\to 2}{lim}\frac{\sqrt{x^2+5}-3}{\sqrt{x^2+14}-4}$ ?

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10. Original problem

    Prove that the derivative of a constant is 0 at all points on its graph (Hint: You must identify the difference between “approaches” and “equals”).

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11. Added problem

    Find ${\lim }_{x\to 2}(x^2+2x-1)$.

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12. Added problem

    Find ${\lim }_{x\to 3}(x^2-9)/(x-3)$ and explain why the quotient can be undefined at 3.

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13. Added problem

    Evaluate ${\lim }_{x\to 0}(\sqrt{9+x}-3)/x$.

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14. Added problem

    Near 1, $f\left(x\right)=x+2$ for $x<1$ and $f\left(x\right)=2x+3$ for $x>1$. Find both one-sided limits and the two-sided limit.

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15. Added problem

    Define $f\left(x\right)=(x^2-4)/(x-2)$ for $x\ne 2$ and $f\left(2\right)=k$. Choose k to make f continuous at 2.

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16. Added problem

    For $f\left(x\right)=3x-2$, give a δ in terms of a prescribed $\epsilon >0$ that proves the limit at any real a.

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17. Added problem

    Prove ${\lim }_{x\to 2}{x}^2=4$ with an explicit epsilon–delta choice.

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18. Added problem

    Evaluate ${\lim }_{x\to 0}\sin \left(5x\right)/\left(2x\right)$, with angles in radians.

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19. Added problem

    Use identities rather than derivatives to evaluate ${\lim }_{x\to 0}(1-\cos (3x\left)\right)/x^2$.

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20. Added problem

    Find ${\lim }_{x\to 0}{x}^2\sin (1/x)$ and justify it without graphing.

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21. Added problem

    Evaluate ${\lim }_{x\to \infty }(6x^2-x)/(2-4x^2)$.

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22. Added problem

    Find ${\lim }_{x\to -\infty }(2x+1)/(x^3-7)$.

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23. Added problem

    Find ${\lim }_{x\to 2^{-}}1/(x-2)$ and ${\lim }_{x\to 2^{+}}1/(x-2)$.

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24. Added problem

    Does ${\lim }_{x\to 0}\left|x\right|/x$ exist? Show the two sides.

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25. Added problem

    A table near $x=2$ gives $f\left(1.9\right)=2.9$, $f\left(1.99\right)=2.99$, $f\left(2.01\right)=3.01$, and $f\left(2.1\right)=3.1$. What limit is suggested, and can the table prove it?

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26. Added problem

    Evaluate ${\lim }_{x\to 2}\sqrt{x^2+1}$ and explain the composition step.

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27. Added problem

    Evaluate ${\lim }_{h\to 0}\left(\right(3+h{)}^2-9)/h$.

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28. Added problem

    Evaluate ${\lim }_{x\to 1}(x^2-1)/(x^2+x-2)$.

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29. Added problem

    Use limits to explain why $f\left(x\right)=\left|x\right|$ is continuous at zero but has no derivative there.

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30. Added problem

    Evaluate ${\lim }_{n\to \infty }(1+3/n{)}^n$ using the exponential special limit.

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