1.
\(\lim\limits_{x\to 5}{\frac{x^{2}+4x-45}{x^{2}-4x-5}}\)
Exercises 1.7 Review exercises
Exercise Group.
Evaluate the limit.
2.
\(\lim\limits_{x\to 7}{\frac{x^{2}-10x+21}{x^{2}-15x+56}}\)
3.
\(\lim\limits_{x\to 9}{\frac{x^{2}+9x+20}{x^{2}+7x+12}}\)
Exercise Group.
Approximate the limit numerically.
4.
\(\lim\limits_{x\to 0.6}{\frac{x^{2}+8.8x-5.64}{x^{2}+6.3x-4.14}}\)
5.
\(\lim\limits_{x\to 1.7}{\frac{x^{2}+4.3x-10.2}{x^{2}-2.9x+2.04}}\)
6.
Evaluate the given limits of the piecewise defined function.
\(f(x) ={\begin{cases}\displaystyle{x^{2}-3}\amp \text{if}\ x \lt 1\cr
\displaystyle{x}\amp \text{if}\ x \ge 1\end{cases}}\)
(a)
\(\lim\limits_{x\to 1^-} f(x)\)
(b)
\(\lim\limits_{x\to 1^+} f(x)\)
(c)
\(\lim\limits_{x\to 1} f(x)\)
(d)
\(f(1)\)
7.
Numerically approximate the following limits.
(a)
\(\lim\limits_{x\to 4.6^{+}}\left({\frac{x^{2}-2.6x-9.2}{x^{2}-9.6x+23}}\right)\)
(b)
\(\lim\limits_{x\to 4.6^{-}}\left({\frac{x^{2}-2.6x-9.2}{x^{2}-9.6x+23}}\right)\)
8.
Give an example of a function \(f\) for which \(\lim\limits_{x\to 0} f(x)\) does not exist.
9.
Use an \(\varepsilon\)-\(\delta\) proof to prove that \(\lim\limits_{x\to 1}(5x-2)=3\text{.}\)
10.
Let \(\lim\limits_{x\to0} f(x) = 1\) and \(\lim\limits_{x\to0} g(x) = -1\text{.}\) Evaluate the following limits.
(a)
\(\lim\limits_{x\to0}(f+g)(x)\)
(b)
\(\lim\limits_{x\to0}(fg)(x)\)
(c)
\(\lim\limits_{x\to0}(f/g)(x)\)
(d)
\(\lim\limits_{x\to0}f(x)^{g(x)}\)
11.
Let \(f(x) = \begin{cases}{-x^{2}-4x-3}\amp x\lt-3\\{x^{2}+7x+14}\amp x\geq-3\end{cases}\text{.}\)
Is \(f\) continuous everywhere?
12.
Find \(\lim\limits_{x\to e} \ln(x)\text{.}\)
13.
Approximate \(\displaystyle\lim_{x\to 2.5}{\frac{x^{2}+\left(-9.5\right)x+17.5}{x^{2}+\left(-1.5\right)x+\left(-2.5\right)}}\text{.}\)
14.
Use the Bisection Method to approximate, accurate to two decimal places, the root of \(g(x) = {x^{3}+4x^{2}+6x+\left(-3\right)}\) on \([0.3,0.4]\text{.}\)
15.
Give maximal intervals on which each of the following functions are continuous.
(a)
\(\dfrac{1}{e^x+1}\)
(b)
\({\frac{1}{x^{2}-1}}\)
(c)
\({\sqrt{11-x}}\)
(d)
\({\sqrt{11-x^{2}}}\)
16.
Use the graph of \(f(x)\) provided to answer the following.
(a)
\(\lim\limits_{x\to-3^-} f(x)\)
(b)
\(\lim\limits_{x\to-3^+} f(x)\)
(c)
\(\lim\limits_{x\to-3} f(x)\)
(d)
Where is \(f\) continuous?
