2.1.4 Practice set: limits

Limits

Limits (lesson 2.1.1)

0 of 6 correct

1.
Find $\displaystyle\lim_{x \to 2} (3x^2 - x)$
2.
Find $\displaystyle\lim_{x \to 5} \frac{x^2 - 25}{x - 5}$
3.
Find $\displaystyle\lim_{x \to 1} \frac{x^2 + x - 2}{x - 1}$
4.
Find $\displaystyle\lim_{x \to 16} \frac{x - 16}{\sqrt{x} - 4}$
5.
Find $\displaystyle\lim_{h \to 0} \frac{(2 - h)^2 - 4}{h}$
6.
$g(x) = x^2$ for every $x \neq 3$ , and $g(3) = 0$ . Find $\displaystyle\lim_{x \to 3} g(x)$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

One-sided limits and continuity

One-sided limits and continuity (lesson 2.1.2)

0 of 7 correct

1.
For $\frac{|x|}{x}$ , what is the limit from the right at 0?
2.
$f(x) = 2x + 1$ for $x < 1$ and $5 - x$ for $x \ge 1$ . Find $\displaystyle\lim_{x \to 1^-} f(x)$
3.
For the same $f$ , what is $f(1)$ ?
4.
$f(x) = 2x + k$ for $x < 3$ and $x^2$ for $x \ge 3$ . Which $k$ makes $f$ continuous?
5.
Find $\displaystyle\lim_{x \to 2^-} \frac{x^2 - 4}{|x - 2|}$
6.
$\frac{x - 2}{x^2 - 4}$ has a hole at 2. What is its limit there?
7.
$f(x) = \frac{1}{x - 1}$ has $f(0) = -1$ and $f(3) = 0.5$ . Bisection on $[0, 3]$ closes in on which input?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Limits at infinity

Limits at infinity (lesson 2.1.3)

0 of 6 correct

1.
Find $\displaystyle\lim_{x \to \infty} \frac{7x + 2}{2x - 1}$
2.
Find $\displaystyle\lim_{x \to \infty} \frac{5x + 3}{x^3 - 2}$
3.
Find $\displaystyle\lim_{x \to -\infty} \frac{1 - 6x^2}{2x^2 + 5}$
4.
Find $\displaystyle\lim_{x \to \infty} \frac{3e^x + x^5}{e^x}$
5.
Find $\displaystyle\lim_{x \to \infty} \frac{x + 500}{5x}$
6.
Find $\displaystyle\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^n$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Does the limit exist?

Write one_sided(f, a, side, h=1e-6) and limit_exists(f, a, tol=1e-4).

one_sided returns f(a - h) when side is "left" and f(a + h) when it is "right": an estimate of each one-sided limit.
limit_exists returns True when the two one-sided estimates are within tol of each other, and False otherwise.

Never call f(a) itself: at a hole it is undefined. The tests use $\frac{x^2 - 1}{x - 1}$ and $\frac{\sin x}{x}$ , whose limits exist, and $x^3$ at 2, where the function is continuous. They also use the jump $\frac{|x|}{x}$ , the piecewise jump from lesson 2.1.2, and the blow-up $\frac{1}{x}$ , which have no limit.

A numerical test can only suggest a limit (lesson 2.1.1). Yours will be fooled by $\frac{1}{x^2}$ at 0: both sides give $10^{12}$ , so it says True, though both sides blow up.

See the tests

import math


def _hole(x):
    return (x**2 - 1) / (x - 1)


def _jump(x):
    return abs(x) / x


def _pieces(x):
    return x**2 if x < 2 else x - 1


def test_left_and_right_estimates():
    """Both sides of (x² − 1)/(x − 1) at 1 are close to 2"""
    expect_close(one_sided(_hole, 1, "left"), 2.0, what="left estimate", rel=1e-5)
    expect_close(one_sided(_hole, 1, "right"), 2.0, what="right estimate", rel=1e-5)


def test_jump_sides():
    """|x|/x is −1 just left of 0 and 1 just right of it"""
    expect_close(one_sided(_jump, 0, "left"), -1.0, what="left of 0")
    expect_close(one_sided(_jump, 0, "right"), 1.0, what="right of 0")


def test_hole_limit_exists():
    """The limit of (x² − 1)/(x − 1) at 1 exists"""
    assert limit_exists(_hole, 1) is True, "expected True for (x² − 1)/(x − 1) at 1"


def test_sin_limit_exists():
    """The limit of sin(x)/x at 0 exists"""
    assert limit_exists(lambda x: math.sin(x) / x, 0) is True, "expected True for sin(x)/x at 0"


def test_continuous_limit_exists():
    """The limit of x³ at 2 exists"""
    assert limit_exists(lambda x: x**3, 2) is True, "expected True for x³ at 2"


def test_jump_limit_does_not_exist():
    """The limit of |x|/x at 0 does not exist"""
    assert limit_exists(_jump, 0) is False, "expected False for |x|/x at 0"


def test_piecewise_jump():
    """x² for x < 2 and x − 1 for x ≥ 2 has no limit at 2 (the sides head for 4 and 1)"""
    expect_close(one_sided(_pieces, 2, "left"), 4.0, what="left of 2", rel=1e-5)
    expect_close(one_sided(_pieces, 2, "right"), 1.0, what="right of 2", rel=1e-5)
    assert limit_exists(_pieces, 2) is False, "expected False for the piecewise jump at 2"


def test_blow_up_does_not_exist():
    """1/x has no limit at 0 (it heads for −∞ on the left and ∞ on the right)"""
    assert limit_exists(lambda x: 1 / x, 0) is False, "expected False for 1/x at 0"

Limits

One-sided limits and continuity

Limits at infinity

Coding exercise

Free further reading