Limits and Continuity (23 Multiple Choice)

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10fehyrf

Mathematics

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1.

Using the graph of f(x) below, find limit as x approaches 3 of f of x

A graph is shown beginning at the open point negative two comma negative four continuing to the open point negative one negative one up to a maximum at zero comma zero and back down to the open point one comma negative one. The graph begins again at the closed point one comma two and then continues down to infinity along the asymptote x equals three then from negative infinity along the asymptote of x equals three the graph increases to the closed point five comma zero. A noncontinuous closed point exists at negative one comma negative two. (4 points)

2.

Find limit as x approaches zero of the quotient of sine of negative 3x and the sine of 2x . (4 points)

3.

What is limits as x approaches negative 4 from the right of the quotient of x and the quantity x plus 4. ? (4 points)

4.

What is limit as x approaches 25 of the quotient of the quotient of the quantity 25 minus x and the square root of x minus 5. ? (4 points)

5.

Find the limit of the function by using direct substitution

limit as x approaches zero of quantity x squared minus five. (4 points)

1.

If limit as x approaches a of f of x equals two and limit as x approaches a of g of x equals four , then limit as x approaches a of the quantity three times f of x squared minus four times g of x equals negative four . (4 points)

2.

Find the limit of the function algebraically.

limit as x approaches five of quantity x squared minus twenty five divided by quantity x minus five. (4 points)

3.

Find limit as x approaches one of f of x for f of x equals the piecewise function 3 x minus 1 when x is less than or equal to one or 2 x squared when x is greater than one . (4 points)

4.

Evaluate limit as x goes to 0 of the quotient of 1 minus cosine squared x and x . (4 points)

5.

Evaluate limit as x goes to 0 of the quotient of the quantity x raised to the 8th power minus 1 and x minus 1 . (4 points)

1.

Evaluate limit as x goes to infinity of the quotient of 4 times x cubed plus 2 times x squared plus 3 times x and negative 9 times x squared plus 5 times x plus 5 (6 points)

2.

Find the equation of the horizontal asymptote for the function, f of x equals the quotient of the quantity x minus 100 and x raised to the 2nd power minus 100 . (7 points)

3.

Which of the following is false for f of x equals the quotient of 10 times x cubed minus 10 times x squared minus 10 times x and the quantity 2 times x raised to the fifth power minus 2 times x ? (7 points)

1.

To two decimal places, find the value of k that will make the function f(x) continuous everywhere.

f of x equals the quantity 3 times x plus k for x less than or equal to 3 and is equal to k times x squared minus 6 for x greater than 3 (4 points)

2.

Where is f of x equals the quotient of x plus 2 and the quantity x squared minus 2 times x minus 8 discontinuous? (4 points)

3.

Is the function f of x equals the quantity 2 times x plus 1 for x less than or equal to 4 and is equal to x squared plus 12 for x greater than 4 continuous? (4 points)

4.

List the discontinuities for the function f(x) = cot( 2x over 3 ). (4 points)

5.

Which of the following is true for f of x equals the quotient of the quantity x squared minus 4 and the quantity x minus 2 ? (4 points)

1.

What is the instantaneous slope of y = negative eight over x at x = -3? ( 4 points)

2.

What is the average rate of change of y with respect to x over the interval [-2, 6] for the function y = 5x + 2? (4 points)

3.

What is the slope for the function y = -5x2 + 2 at the point x = 1? (4 points)

4.

The surface area, S, of a sphere of radius r feet is S = S(r) = 4πr2. Find the instantaneous rate of change of the surface area with respect to the radius r at r = 4. (4 points)

5.

A ball is thrown vertically upward from the top of a 100 foot tower, with an initial velocity of 10 ft/sec. Its position function is s(t) = -16t2 + 10t + 100. What is its velocity in ft/sec when t = 2 seconds? (4 points)

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Explanation & Answer

All answers was tested, so please use them. Ask if something is unclear.docx and pdf files are identical

1. B, does not exist. (we can say it is −∞ but no such option).
3

2. D, − 2 (because sin(−3𝑥 ) ~ − 3𝑥, sin(2𝑥 ) ~2𝑥).
3. D, −∞ (𝑥 → −4, 𝑥 + 4 → 0+ ).
25 −𝑥

4. B, -10 (√𝑥 −5 =

(5−√𝑥 )(5+√𝑥)
√𝑥−5

= −(5 + √ 𝑥)).

5. D, -5 (obvious).
1. A, TRUE (that limit is 3 ∙ 22 − 4 ∙ 4 = 12 − 16 = −4).
2. C, 10 (the function is equal to 𝑥 + 5).
3. A, 2 (the same from both sides).
4. C, 0 (1 − cos 2 𝑥 = sin2 𝑥 ,

sin2 𝑥
𝑥

= sin 𝑥 ∙

sin 𝑥
𝑥

→ 0).

5. C, 1 (direct substitution).
1. B, −∞ (the denominat...


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