Roll No Name:
Test #1, September 7th, 2006
Math 1131
1. Determine the following limits numerically and analytically:
(a) lim
x→∞ µx+ 1
x−1¶3x
(b) lim
x→0
1−cos 4x
xsin x(c) lim
x→64
3
√x−4
√x−8
2. Determine if the following functions are continuous or not. If they are not continuous
find the points of discontinuity.
(a)f(x) =
x2+x+ 2 if x ≥0
sin 2x
xif x < 0,(b)g(x) =
ln(e+x) + ln(e2+x)if x ≥0
−ln(1 + 2|x|)
xif x < 0.
3. Find all values of asuch that the following function is continuous:
h(x) =
ax
3 + a2xif x ≥1
4
√x−1
x−1if x < 1
.
4. Determine the constants aand bsuch that the function g(x) =
ax2+bx if x < −3or x ≥2
ax +b+ 1 if x ∈[−3,2)
becomes continuous at every point.
5. Find the derivative of the following functions at every point in their natural domain using
the definition of the derivative.
(a)h(x) = √3x+ 1 (b)i(x) = x−1
2x−1(c)(Bonus)k(x) = x2
2x+ 1
I
