#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.
#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
#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+x-x^2)/(2x+sqrt(4x^4-3)) -> -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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