Additional Practice Questions:
Consider the following graph of \(y=f(x)\) for the questions 1, 2, and 3.
1. Find the following values, if they exist.
(a) \(\displaystyle{\lim_{x \to 0^-} f(x) } \) (b) \(\displaystyle{\lim_{x \to 0^+} f(x) } \) (c) \(\displaystyle{\lim_{x \to 0} f(x) } \)
(d) \(\displaystyle{\lim_{x \to 2^-} f(x) } \) (e) \(\displaystyle{\lim_{x \to 2^+} f(x) } \) (f) \( f(-4)\) (g) \(f'(-3)\)
2. Consider the above graph of \(f\). Is \( f \) continuous for the following values of \( x \)? If you say Yes, justify your answer using the definition of continuity. If you say No, state what type of discontinuity (removable, jump or infinite) occurs at that value of \(x\).
\(x\) value
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Continuous?
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Reason/Type
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-4
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Yes No
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-1
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Yes No
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0
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Yes No
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2
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Yes No
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3. Does \(f\) have a derivative at each of the following \(x\) values? In the table below, circle Yes or No. If you circle No, state the reason, and if you circle Yes, evaluate the derivate.
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\( x \) value
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Differentiable?
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Reason/Derivative
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-5
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Yes No
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-1
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Yes No
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0
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Yes No
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(4) Evaluate the following limits:
solution
(a) \( \displaystyle{\lim_{x \to 1/2} \left(\cos(\pi
x)-e^{x-\frac{1}{x}} \right)} \) (b)
\(\displaystyle{\lim_{x \to -1} \frac{x^2-x-2}{7x+7} } \) (c)
\(\displaystyle{\lim_{h \to 0} \frac{\sqrt{16+h}-4}{h} } \)
(d) \(\displaystyle{\lim_{x \to 0} \dfrac{\frac{1}{x+4}-\frac{1}{4}}{x} } \)
(e) \(\displaystyle{\lim_{x \to \infty} \frac{3x^5+5x+2}{2x^5-2x^3+1} } \)
(4.5) Evaluate the limits or state that they do not exist.
solution
(a) \( \displaystyle{\lim_{x \to 2^-} \frac{|x-2|}{(2x-4)}} \)
(b) \( \displaystyle{\lim_{x \to 2^+} \frac{|x-2|}{(2x-4)}} \) (c) \(
\displaystyle{\lim_{x \to 2} \frac{|x-2|}{(2x-4)}} \)
(5)
Find an equation of the tangent line to the curve \( y=\sqrt{x} \)
at \( x=1 \). Write your answer in the slope-intercept form: \(
y=mx+b\). Note that the derivative of \( y=\sqrt{x} \) is
given by \( y'=\frac {1}{2\sqrt{x}} \).
6. Find the limit if it exists.
(a) \(\displaystyle{\lim_{x \to \infty} e^{x-x^2} } \) (b) \(\displaystyle{\lim_{x \to 1} \frac{1}{x-1} } \)
7. Determine whether each statement below is true or false. If the statement is true briefly explain why.
If the statement is false, provide a counterexample or sketch a graph.
(a) If both the left-hand limit and right-hand limit of a function at a number exist, then the function
is continuous at that number.
(b) If a function is continuous at a number then it is differentiable at that number.
(c) If a function is discontinuous at a number then the function is undefined at that number.
8.
(a) I and II (b) II and III (c) I and III (d) III only (e) all