Evaluate \(\lim_{x\to 3} (x+3)\).
6
Evaluate \(\lim_{x\to \pi} (x - \tfrac{22}{7})\).
\(\pi - \tfrac{22}{7}\)
Evaluate \(\lim_{r\to 1} \pi r^{2}\).
\(\pi\)
Evaluate \(\lim_{x\to 4} \dfrac{4x+3}{x-2}\).
\(\tfrac{19}{2}\)
Evaluate \(\lim_{x\to 1} \dfrac{x^{10}+x^{5}+1}{x-1}\).
\(-\tfrac{1}{2}\)
Evaluate \(\lim_{x\to 0} \dfrac{(x+1)^{5}-1}{x}\).
5
Evaluate \(\lim_{x\to 2} \dfrac{3x^{2}-x-10}{x^{2}-4}\).
\(\tfrac{11}{4}\)
Evaluate \(\lim_{x\to 3} \dfrac{x^{4}-81}{2x^{2}-5x-3}\).
\(\tfrac{108}{7}\)
Evaluate \(\lim_{x\to 0} \dfrac{ax+b}{cx+1}\).
\(b\)
Evaluate \(\lim_{z\to 1} \dfrac{z^{1/3}-1}{z^{1/6}-1}\).
2
Evaluate \(\lim_{x\to 1} \dfrac{ax^{2}+bx+c}{cx^{2}+bx+a}\), where \(a+b+c\neq 0\).
1
Evaluate \(\lim_{x\to -2} \dfrac{\tfrac{1}{x}+\tfrac{1}{2}}{x+2}\).
\(-\tfrac{1}{4}\)
Evaluate \(\lim_{x\to 0} \dfrac{\sin ax}{bx}\).
\(\tfrac{a}{b}\)
Evaluate \(\lim_{x\to 0} \dfrac{\sin ax}{\sin bx}\), where \(a,b\neq 0\).
\(\tfrac{a}{b}\)
Evaluate \(\lim_{x\to \pi} \dfrac{\sin(\pi - x)}{\pi(\pi - x)}\).
\(\tfrac{1}{\pi}\)
Evaluate \(\lim_{x\to 0} \dfrac{\cos x}{\pi - x}\).
\(\tfrac{1}{\pi}\)
Evaluate \(\lim_{x\to 0} \dfrac{\cos 2x - 1}{\cos x - 1}\).
4
Evaluate \(\lim_{x\to 0} \dfrac{ax + x\cos x}{b\sin x}\).
\(\tfrac{a+1}{b}\)
Evaluate \(\lim_{x\to 0} x\sec x\).
0
Evaluate \(\lim_{x\to 0} \dfrac{\sin ax + bx}{ax + \sin bx}\), where \(a, b, a+b \neq 0\).
1
Evaluate \(\lim_{x\to 0} (\csc x - \cot x)\).
0
Evaluate \(\lim_{x\to \pi/2} \dfrac{\tan 2x}{x - \pi/2}\).
2
Find \(\lim_{x\to 0} f(x)\) and \(\lim_{x\to 1} f(x)\), where
\(f(x) = \begin{cases} 2x+3, & x \le 0 \\ 3(x+1), & x>0 \end{cases}\).
\(\lim_{x\to 0} f(x) = 3\)
\(\lim_{x\to 1} f(x) = 6\)
Find \(\lim_{x\to 1} f(x)\), where
\(f(x) = \begin{cases} x^{2}-1, & x \le 1 \\ -x^{2}-1, & x>1 \end{cases}\).
Limit does not exist at \(x=1\).
Evaluate \(\lim_{x\to 0} f(x)\), where
\(f(x) = \begin{cases} \dfrac{|x|}{x}, & x \ne 0 \\ 0, & x=0 \end{cases}\).
Limit does not exist at \(x=0\).
Find \(\lim_{x\to 0} f(x)\), where
\(f(x) = \begin{cases} \dfrac{x}{|x|}, & x \ne 0 \\ 0, & x=0 \end{cases}\).
Limit does not exist at \(x=0\).
Find \(\lim_{x\to 5} f(x)\), where \(f(x) = |x| - 5\).
0
Suppose
\(f(x) = \begin{cases} a+bx, & x<1 \\ 4, & x=1 \\ b-ax, & x>1 \end{cases}\)
and if \(\lim_{x\to 1} f(x) = f(1)\), what are possible values of \(a\) and \(b\)?
\(a=0,\ b=4\)
Let \(a_{1}, a_{2}, \ldots, a_{n}\) be fixed real numbers and define
\(f(x) = (x-a_{1})(x-a_{2})\cdots(x-a_{n})\).
What is \(\lim_{x\to a_{1}} f(x)\)? For some \(a \ne a_{1}, a_{2}, \ldots, a_{n}\), compute \(\lim_{x\to a} f(x)\).
\(\lim_{x\to a_{1}} f(x) = 0\)
\(\lim_{x\to a} f(x) = (a-a_{1})(a-a_{2})\cdots(a-a_{n})\)
If
\(f(x) = \begin{cases} |x|+1, & x<0 \\ 0, & x=0 \\ |x|-1, & x>0 \end{cases}\),
for what value(s) of \(a\) does \(\lim_{x\to a} f(x)\) exist?
\(\lim_{x\to a} f(x)\) exists for all \(a \ne 0\).
If the function \(f(x)\) satisfies
\(\displaystyle \lim_{x\to 1} \dfrac{f(x) - 2}{x^{2} - 1} = \pi\),
evaluate \(\lim_{x\to 1} f(x)\).
2
If
\(f(x) = \begin{cases} mx^{2}+n, & x<0 \\ nx+m, & 0 \le x \le 1 \\ nx^{3}+m, & x>1 \end{cases}\).
For what integers \(m\) and \(n\) does both \(\lim_{x\to 0} f(x)\) and \(\lim_{x\to 1} f(x)\) exist?
For \(\lim_{x\to 0} f(x)\) to exist, we need \(m=n\); \(\lim_{x\to 1} f(x)\) exists for any integral values of \(m\) and \(n\).