MATH 21A

Midterm 1 | Fall '15 | Hunter

[20%] Say if the following statements are true of false. (For the question only, you don't have to explain your answers).
(a) If lim$_{x \to c^+}f(x)=$3 and lim$_{x \to c^-}f(x)=$3.001, then lim$_{x \to c} f(x)$ is close to 3. False. (If the left and right limits are different, then the limit does not exist.)

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(b) If lim$_{x \to c}$exists, then $f(x)$ is continuous at $c$.

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(c) If $f(x)$ is continuous at $c$, then lim$_{x\to c}f(x)$ exists.

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(d) If $\lim_{x \to c}f(x) = 0$, then $f(x) > 0$ for all $x > 0$ that are sufficiently close to $0$.

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[30%] Evaluate the following limits or say if they do not exist.
(a) $\lim_{x \to 2} \frac{2x^2 + 1}{11 - x^3}$

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(b) $\lim_{x \to 0} \frac{\sqrt{5x + 4} - 2}{x}$

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(c) $\lim_{x \to 0} \frac{\sin(1/x)}{x}$

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(d) $\lim_{x \to 3^-} \frac{x^2 - 2x - 3}{|x - 3|}$

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[15%] Evaluate the following limits involving infinity.
(a) $\lim_{x \to 1} \frac{x + 1}{(x - 1)^2(x - 2)}$

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(b) $\lim_{x \to \infty} \sqrt{\frac{x^2 + 2x + 4}{4x^2 + 2x + 1}}$

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(c) $\lim_{x \to -\infty} xe^{1/x}$

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[20%] Define a function f($x$) for all real numbers except $x$ = −1, 0, 2 by


(a) Evaluate $\lim_{x \to 1^-}$ and $\lim_{x \to 1^+}f(x)$. Can you choose a value of $f(−1)$ so that $f(x)$ is continuous at $x = −1$; if so, what is $f(−1)$?

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(b) Evaluate $\lim_{x \to 0^-} f(x)$ and $\lim_{x \to 0^+} f(x)$. Can you choose a value of $f(0)$ so that $f(x)$ is continuous at $x = 0$; if so, what is $f(0)$?

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(c) Evaluate $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$. Can you choose a value of $f(2)$ so that $f(x)$ is continuous at $x = 2$; if so, what is $f(2)$?

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5. [15%] (a) Suppose that a function $f(x)$ is defined for all $x$ in an interval about $c$, except possibly at $c$ itself. Give the precise $\varepsilon$-$\delta$ definition of



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(b) Use the $\varepsilon$-$\delta$ definition to prove that


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