Using the fact that \(\lim_{x \to 0} \sin(x)/x = 1\)
and the “limit laws" demonstrate how to algebraically find
the values of the following limits.
\[
\lim\limits_{x \to 0} \frac{\sin(3x)}{7x}
\qquad
\lim\limits_{x \to 0} \frac{\sin(3x)}{\sin(7x)}
\qquad
\lim\limits_{x \to 0} \frac{x\cos(x)}{\tan(x)}
\qquad
\lim\limits_{x \to 0} \frac{\sin^2(2x)}{x^2}
\]
Without referencing technology,
determine the value of each limit.
\[
\lim\limits_{x \to \infty} \sqrt{x}
\qquad
\lim\limits_{x \to 7^+} \frac{x^2-x-40}{x-7}
\qquad
\lim\limits_{x \to 7^-} \frac{x^2-3x-28}{x-7}
\qquad
\lim\limits_{x \to 7} \frac{49}{(x-7)^2}
\]
\[
\lim\limits_{x \to \infty} \frac{3x^2-7x+5}{2x-1}
\qquad
\lim\limits_{x \to \infty} \frac{3x^2-7x+5}{(2x-1)^2}
\qquad
\lim\limits_{x \to \infty} \frac{3x^2-7x+5}{(2x-1)^3}
\]
Demonstrate how to evaluate each of these limits.
Hint: “conjugate”
\[
\lim\limits_{x \to 1} \frac{1-\sqrt{x}}{1-x}
\qquad
\lim\limits_{x \to 0} \frac{1-\sqrt{1-x^2}}{x}
\qquad
\lim\limits_{x \to 0} \frac{1-\sqrt{1-x^2}}{x^2}
\]
Let \(g\) be the function defined piecewise as
\[
g(x) = \begin{cases}
-x &\text{for } x \lt 1 \qquad &7 &\text{for } x=1
\\x^2-2 &\text{for } 1 \lt x \leq 5 \qquad &7x-5 &\text{for } 5 \lt x
\end{cases}
\]
Evaluate the following. If a limit doesn’t exist, say so.
\( \displaystyle \lim\limits_{x \to 1^-} g(x)\)
\( \displaystyle \lim\limits_{x \to 1} g(x)\)
\( \displaystyle g(1)\)
\( \displaystyle \lim\limits_{x \to 5^-} g(x)\)
\( \displaystyle \lim\limits_{x \to 5^+} g(x)\)
\( \displaystyle \lim\limits_{x \to 5} g(x)\)
Use a computing device to approximate
the value of these limits
accurate to seven decimal places.
\[
\lim\limits_{x \to \infty} x^{\frac{1}{x}}
\qquad
\lim\limits_{x \to 0} \; 4\arctan\!\left(x^x\right)
\qquad
\lim\limits_{x \to \infty} \left(\frac{x}{x+1}\right)^x
\]
For functions \(f\) and \(g,\)
it may be that \(\lim_{x \to c}\big(f(x) + g(x)\big)\) exists
even though neither \(\lim_{x \to c}f(x)\) nor \(\lim_{x \to c}g(x)\) exist.
Find examples of such functions.
For functions \(f\) and \(g,\)
it may be that \(\lim_{x \to c}\big(f(x)g(x)\big)\) exists
even though neither \(\lim_{x \to c}f(x)\) nor \(\lim_{x \to c}g(x)\) exist.
Find examples of such functions.
Let \(g\) be the function defined piecewise as
\[
g(x) = \begin{cases}
-1 &\text{ for } x \lt -3
\\f(x) &\text{ for } -3 \leq x \lt 1
\\-x^2+5 &\text{ for } x \geq 1
\end{cases}
\]
Find an example of a linear function \(f\) such that \(g\) is continuous.
Find an example of a non-linear function \(f\) such that \(g\) is continuous.
How can you very quickly prove
that the polynomial \(x^{171}-7x^2-1\)
has a root on the domain \([0,2]\)
by invoking the Intermediate Value Theorem?
Challenges
Patrick Snyder
Demonstrate how to compute the following limit algebraically.
\[ \lim_{x\ to 0} \frac{\sqrt{1+\tan(x)}-\sqrt{1+\sin(x)}}{x^3} \]
James Stewart
Does there exist a number
that is exactly one more than its cube?
The challenge is to cunningly answer this question
without calculating any such number.
If \(\lim_{x \to c} \big(f(x) + g(x)\big) = 11\)
and \(\lim_{x \to c} \big(f(x) - g(x)\big) = 7,\)
what must the value of
\(\lim_{x \to c} \big(f(x)g(x)\big)\) be?
True or False?
Suppose that a function \(f\) defined on some domain \([a,b]\)
has the following property:
for any \(N\) between \(f(a)\) and \(f(b)\)
there exists some \(c\) between \(a\) and \(b\)
such that \(f(c) = N.\)
Then \(f\) must be a continuous function.
