mat
Practice problems for Midterm 1
Math 17A, Winter 2026
1. Consider the functions:
This practice sheet contains more problems than the actual exam.
x2
f (x) = x2 + 1
x→ 1 , h(x) = 1
→ 1 , g(x) = ↑x
ln x
.
a) Find the domain
For each of them:
c) Sketch the graph
b) Find all vertical and horizontal asymptotes (if any)
f (x) = sin x, if x < c,
2. For which c the function
is continuous?
cos x, if x ↓ c
a)
3. Compute the limits:
lim
b)
↑x + 1
x→0(sin3 x→ cos x)
lim
c)
x2
x→0
→ 2x→ 3
↑x
x→3
lim
x2
.
d)
→ 3x
lim
7x3 + 8x2 + 9x
e)
x→0
→ 13x3.
11x + 12x2
x→+↑
lim
f)
e↓x2
x→+↑
lim
1
(ln(x + 1)→ ln x)
7x2 + 8x + 9
g)
x→+↑
lim
11 + 12x→ 13x2.
h)
lim
2x3 + ↑x4 + 7
3x3
x→+↑
4. Find limx→1 f (x), where
→ 5
f (x) = x2 + 3x + 4, if x ↓ 1,
5. Plot the graph of the function
sgn(x) =
7x2 + 1, if x < 1.
x , if x ↔= 0
|x|
Is it continuous?
0, if x = 0.
f (x) = x2 + x→ 9, if |x| ↓ 3,
6. For the function
find one-sided limits at x =→3, x = 0, x = 3. Is this function continuous?
x, if |x| < 3,
x3 + x→ 3 = 0
7*. Use the Intermediate Value Theorem to show that the equation
2
has a solution on the interval [1, 2].