MATH 140 Section 01** Sample Exam 1
1. Use the secant-line/limit approach to find the slope of the line tangent to f (x) = 2x2 + x − 1 at
x = α, where α is a fixed but unknown number. Use full sentences to clarify your mathematics.
2. For the following graph of a function j(x), categorize all limits lim j(x) for all real numbers
x→a
a. You should do left and right-hand limits only if they differ. You can reference j(a) for some
a only if it exists.
3. Evaluate each of the following. Each answer should either be a number, +∞, −∞ or DNE.
You may only use rules given in the course. Substitutions should be explicitly given.
sin(4x)
x→0 sin(3x)
(a) lim
(b) lim−
x2 −5x+6
|x−2|
(c) lim+
x+7
|x−2|
x→2
x→2
2
1+ x
1
3−
x→0
x
(d) lim
(e) lim x2 cos
x→0
(f)
lim
x→π/2−
1
x
x sec x
(g) lim ⌊x⌋, where ⌊x⌋ denotes the greatest integer less than or equal to x and a is a fixed
x→a−
but unknown integer.
4. Find and justify with limits all vertical asymptotes for the function f (x) =
|x+2|
x2 +7x+10
5. Show that each of the following functions is or is not continuous at the point given.
(
x2 − 4x + 1 if x ≥ 1
(a) f (x) =
is continuous from the right at x = 1.
x−5
if x < 1
(b) f (x) from above is not continuous from the left at x = 1.
(c) h(x) =
|x|
x
is not continuous at x = 0.