Check my CALCULUS work, please! :)
posted by Samantha .
Question 1.
lim h>0(sqrt 49+h7)/h =
14
1/14***
0
7
1/7
Question 2.
lim x>infinity(12+x3x^2)/(x^24)=
3***
2
0
2
3
Question 3.
lim x>infinity (5x^3+x^7)/(e^x)=
infinity***
0
1
3
Question 4. Given that:
x 6.8 6.9 6.99 7.01 7.1 7.2
g(x) 9.44 10.21 10.92 11.08 11.31 12.56
it would appear that lim x>7 g(x)=
0
7
11***
The limit does not exist.
x + 4
Question 5. Let f be defined as follows, where a ≠ 0,
f(x)={(x^22ax+a^2)/(xa) if x ≠ a
{5 if x=a
Which of the following are true about f ?
I. lim f(x) x> a exists.
II. f(a) exists.
III. f(x) is continuous at x=a.
None
I, II, and III
I only
II only
I and II only***
Question 7. lim x> infinity (2+xx^2)/(2x+sqrt(4x^43)) =
infinity
0
1***
4
1/2
Question 8. lim x>0+ (cosx/x)=
0
1/2
1
sqrt(2)/2
The limit does not exist.***
Question 9. If g(x) is continuous for all real numbers and g(3) = −1, g(4) = 2, which of the following are necessarily true?
I. g(x)=1 at least once.
II. lim x>3.5 g(x)=g(3.5)
III. lim x>3 g(x) = lim x>3+ g(x)
I only
II only
I and II only
I, II, and III
None of these
Question 10. If the following function is continuous, what is the value of a + b?
h(t){3t^22t + 1, if t<0
{acos(t)+b, if 0≤t≤pi/3
{4sin^2t, if t>pi/3
0
1
2
3
4

#1 and #2 are correct
#3  here is a neat trick that works for most limit questions.
Use your calculator and try a number close to your approach value
e.g. for #1 I used h = .001 and then evaluated the expression
for #3, try a "large" number. However for this one even a good calculator overloads even for relative small "large" numbers
I tried x = 1000 and got ERROR 1 on my calculator
I backed right down to x = 50 , and the result was
appr 1.5 x 10^10
looks very close to zero to me.
#4 all values <7 yield positive number getting close to 11 and values >7 are suddenly negative but close to 11
I would say the limit does not exist.
skipping #5
#7 intuitive approach ...
as x > infinity, √(4x^4  3) > 2x^
so we approach (2 + x  x^2)/(2x + 2x^2
which approaching 1/2
(also try my method of checking with a calculator)
I used x = 1000 and got .499 or close to 1/2
#8 correct 
#3 is 0. exponentials grow much faster than any power of x.
l'Hospital's Rule shows that given enough iterations, the derivatives in the numerator go to zero while that pesky e^x remains in the bottom.
#4. Note that g(x) changes sign, so the limit does not exist. The limit from the left is different from the limit on the right.
#7. As x gets huge, only the highest power matters, so
(2+xx^2)/(2x+sqrt(4x^43)) > x^2/√4x^4 = x^2/2x^2 = 1/2
#9. all are true
#10. h must be continuous at x=0, so since
lim x>0 = 1
acos(0)+b = a+b
we must have
1 = a+b
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