Questions 1, 2, and 3
Consider the following graph of \(y=f(x)\):
Problem 1
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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) \(f'(-3)\)
Problem 2
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Is \( f \) continuous at these values of \( x \)? If No, state the type of discontinuity.
| \(x\) value | Continuous? | Reason / Type |
|---|---|---|
| -4 | Yes | No | |
| -1 | Yes | No | |
| 0 | Yes | No | |
| 2 | Yes | No |
Problem 3
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Is \( f \) differentiable at these values? Evaluate the derivative if Yes.
| \(x\) value | Differentiable? | Reason / Derivative |
|---|---|---|
| -5 | Yes | No | |
| -1 | Yes | No | |
| 0 | Yes | No |
Problem 4
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Evaluate the following limits:
(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) \(\displaystyle{\lim_{x \to \infty} \frac{3x^5+5x+2}{2x^5-2x^3+1} } \)
Problem 5
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Evaluate the limits or state if they do not exist:
(a) \( \displaystyle{\lim_{x \to 2^-} \frac{|x-2|}{(2x-4)}} \)
(b) \( \displaystyle{\lim_{x \to 2^+} \frac{|x-2|}{(2x-4)}} \)
(c) \( \displaystyle{\lim_{x \to 2} \frac{|x-2|}{(2x-4)}} \)
Problem 7
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Find the limit if it exists:
(a) \(\displaystyle{\lim_{x \to \infty} e^{x-x^2} } \)
(b) \(\displaystyle{\lim_{x \to 2^-} \frac{1}{x-2} } \)
(c) \(\displaystyle{\lim_{x \to 1} \frac{1}{x-1} } \)
Problem 8
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Determine True or False for each statement:
(a) If both left/right limits exist, the function is continuous.
(b) If a function is continuous, it is differentiable.
(c) If a function is discontinuous, it is undefined.