Practice Problems
\(\textbf{1)}\) As the graph goes to the right, it approaches \(y=2\). Write the limit notation. The answer is \(\displaystyle \lim_{x\to\infty}f(x)=2\) \(x\to\infty\) means we are looking far to the right.
\(f(x)\to2\) means the y-values approach 2.
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=2\)
\(f(x)\to2\) means the y-values approach 2.
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=2\)
\(\textbf{2)}\) As the graph goes to the left, it goes down without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to-\infty}f(x)=-\infty\) \(x\to-\infty\) means we are looking far to the left.
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-\infty}f(x)=-\infty\)
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-\infty}f(x)=-\infty\)
\(\textbf{3)}\) As \(x\) approaches \(2\) from the left, the graph goes down without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to2^-}f(x)=-\infty\) Approaching 2 from the left is written \(x\to2^-\).
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to2^-}f(x)=-\infty\)
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to2^-}f(x)=-\infty\)
\(\textbf{4)}\) As the graph goes to the left, it goes up without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to-\infty}f(x)=\infty\) \(x\to-\infty\) means we are looking far to the left.
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-\infty}f(x)=\infty\)
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-\infty}f(x)=\infty\)
\(\textbf{5)}\) As the graph goes to the right, it approaches \(y=-4\). Write the limit notation. The answer is \(\displaystyle \lim_{x\to\infty}f(x)=-4\) Going to the right means \(x\to\infty\).
Approaching \(y=-4\) means \(f(x)\to-4\).
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=-4\)
Approaching \(y=-4\) means \(f(x)\to-4\).
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=-4\)
\(\textbf{6)}\) As the graph goes to the right, it goes up without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to\infty}f(x)=\infty\) Going to the right means \(x\to\infty\).
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=\infty\)
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=\infty\)
\(\textbf{7)}\) As the graph goes to the left, it approaches \(y=5\). Write the limit notation. The answer is \(\displaystyle \lim_{x\to-\infty}f(x)=5\) Going to the left means \(x\to-\infty\).
Approaching \(y=5\) means \(f(x)\to5\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-\infty}f(x)=5\)
Approaching \(y=5\) means \(f(x)\to5\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-\infty}f(x)=5\)
\(\textbf{8)}\) As the graph goes to the right, it goes down without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to\infty}f(x)=-\infty\) Going to the right means \(x\to\infty\).
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=-\infty\)
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=-\infty\)
\(\textbf{9)}\) As \(x\) approaches \(3\) from the right, the graph goes up without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to3^+}f(x)=\infty\) Approaching 3 from the right is written \(x\to3^+\).
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to3^+}f(x)=\infty\)
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to3^+}f(x)=\infty\)
\(\textbf{10)}\) As \(x\) approaches \(4\) from the left, the graph goes up without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to4^-}f(x)=\infty\) Approaching 4 from the left is written \(x\to4^-\).
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to4^-}f(x)=\infty\)
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to4^-}f(x)=\infty\)
\(\textbf{11)}\) As \(x\) approaches \(-2\) from the right, the graph goes down without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to-2^+}f(x)=-\infty\) Approaching -2 from the right is written \(x\to-2^+\).
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-2^+}f(x)=-\infty\)
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-2^+}f(x)=-\infty\)
\(\textbf{12)}\) As \(x\) approaches \(1\) from the left, the graph goes down without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to1^-}f(x)=-\infty\) Approaching 1 from the left is written \(x\to1^-\).
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to1^-}f(x)=-\infty\)
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to1^-}f(x)=-\infty\)
\(\textbf{13)}\) As \(x\) approaches \(6\) from the right, \(f(x)\) approaches \(3\). Write the limit notation. The answer is \(\displaystyle \lim_{x\to6^+}f(x)=3\) Approaching 6 from the right is written \(x\to6^+\).
The function values approach \(3\).
\(\,\,\,\,\,\displaystyle \lim_{x\to6^+}f(x)=3\)
The function values approach \(3\).
\(\,\,\,\,\,\displaystyle \lim_{x\to6^+}f(x)=3\)
\(\textbf{14)}\) As \(x\) approaches \(5\) from the left, \(f(x)\) approaches \(7\). Write the limit notation. The answer is \(\displaystyle \lim_{x\to5^-}f(x)=7\) Approaching 5 from the left is written \(x\to5^-\).
The function values approach \(7\).
\(\,\,\,\,\,\displaystyle \lim_{x\to5^-}f(x)=7\)
The function values approach \(7\).
\(\,\,\,\,\,\displaystyle \lim_{x\to5^-}f(x)=7\)
\(\textbf{15)}\) As \(x\) approaches \(-3\) from both sides, \(f(x)\) approaches \(8\). Write the limit notation. The answer is \(\displaystyle \lim_{x\to-3}f(x)=8\) Approaching from both sides means there is no \(+\) or \(–\) superscript.
The function values approach \(8\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-3}f(x)=8\)
The function values approach \(8\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-3}f(x)=8\)
\(\textbf{16)}\) As \(x\) approaches \(2\) from both sides, \(f(x)\) approaches \(-5\). Write the limit notation. The answer is \(\displaystyle \lim_{x\to2}f(x)=-5\) Approaching from both sides is written \(x\to2\).
The function values approach \(-5\).
\(\,\,\,\,\,\displaystyle \lim_{x\to2}f(x)=-5\)
The function values approach \(-5\).
\(\,\,\,\,\,\displaystyle \lim_{x\to2}f(x)=-5\)
\(\textbf{17)}\) As \(x\) approaches \(0\) from the right, the graph goes up without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to0^+}f(x)=\infty\) Approaching 0 from the right is written \(x\to0^+\).
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to0^+}f(x)=\infty\)
Going up without bound means \(f(x)\to\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to0^+}f(x)=\infty\)
\(\textbf{18)}\) As \(x\) approaches \(0\) from the left, the graph goes down without bound. Write the limit notation. The answer is \(\displaystyle \lim_{x\to0^-}f(x)=-\infty\) Approaching 0 from the left is written \(x\to0^-\).
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to0^-}f(x)=-\infty\)
Going down without bound means \(f(x)\to-\infty\).
\(\,\,\,\,\,\displaystyle \lim_{x\to0^-}f(x)=-\infty\)
\(\textbf{19)}\) As the graph goes to the right, it approaches the x-axis. Write the limit notation. The answer is \(\displaystyle \lim_{x\to\infty}f(x)=0\) Going to the right means \(x\to\infty\).
Approaching the x-axis means the y-values approach \(0\).
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=0\)
Approaching the x-axis means the y-values approach \(0\).
\(\,\,\,\,\,\displaystyle \lim_{x\to\infty}f(x)=0\)
\(\textbf{20)}\) As the graph goes to the left, it approaches \(y=-3\). Write the limit notation. The answer is \(\displaystyle \lim_{x\to-\infty}f(x)=-3\) Going to the left means \(x\to-\infty\).
Approaching \(y=-3\) means \(f(x)\to-3\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-\infty}f(x)=-3\)
Approaching \(y=-3\) means \(f(x)\to-3\).
\(\,\,\,\,\,\displaystyle \lim_{x\to-\infty}f(x)=-3\)
\(\textbf{21)}\) \(\displaystyle \lim_{x\to\infty}f(x)=6\), Describe the limit in words. The answer is: As the graph goes to the right, it approaches \(y=6\). \(x\to\infty\) means the graph is moving far to the right.
\(f(x)\to6\) means the y-values approach 6.
As the graph goes to the right, it approaches \(y=6\).
\(f(x)\to6\) means the y-values approach 6.
As the graph goes to the right, it approaches \(y=6\).
\(\textbf{22)}\) \(\displaystyle \lim_{x\to-\infty}f(x)=-2\), Describe the limit in words. The answer is: As the graph goes to the left, it approaches \(y=-2\). \(x\to-\infty\) means the graph is moving far to the left.
\(f(x)\to-2\) means the y-values approach -2.
As the graph goes to the left, it approaches \(y=-2\).
\(f(x)\to-2\) means the y-values approach -2.
As the graph goes to the left, it approaches \(y=-2\).
\(\textbf{23)}\) \(\displaystyle \lim_{x\to5^-}f(x)=\infty\), Describe the limit in words. The answer is: As \(x\) approaches \(5\) from the left, the graph goes up without bound. \(5^-\) means approach 5 from the left.
\(\infty\) means \(f(x)\) goes up without bound.
As \(x\) approaches \(5\) from the left, the graph goes up without bound.
\(\infty\) means \(f(x)\) goes up without bound.
As \(x\) approaches \(5\) from the left, the graph goes up without bound.
\(\textbf{24)}\) \(\displaystyle \lim_{x\to5^+}f(x)=-\infty\), Describe the limit in words. The answer is: As \(x\) approaches \(5\) from the right, the graph goes down without bound. \(5^+\) means approach 5 from the right.
\(-\infty\) means \(f(x)\) goes down without bound.
As \(x\) approaches \(5\) from the right, the graph goes down without bound.
\(-\infty\) means \(f(x)\) goes down without bound.
As \(x\) approaches \(5\) from the right, the graph goes down without bound.
\(\textbf{25)}\) \(\displaystyle \lim_{x\to-4^-}f(x)=3\), Describe the limit in words. The answer is: As \(x\) approaches \(-4\) from the left, \(f(x)\) approaches \(3\). \(-4^-\) means approach -4 from the left.
The value \(3\) tells us what \(f(x)\) approaches.
As \(x\) approaches \(-4\) from the left, \(f(x)\) approaches \(3\).
The value \(3\) tells us what \(f(x)\) approaches.
As \(x\) approaches \(-4\) from the left, \(f(x)\) approaches \(3\).
\(\textbf{26)}\) \(\displaystyle \lim_{x\to-4^+}f(x)=3\), Describe the limit in words. The answer is: As \(x\) approaches \(-4\) from the right, \(f(x)\) approaches \(3\). \(-4^+\) means approach -4 from the right.
The value \(3\) tells us what \(f(x)\) approaches.
As \(x\) approaches \(-4\) from the right, \(f(x)\) approaches \(3\).
The value \(3\) tells us what \(f(x)\) approaches.
As \(x\) approaches \(-4\) from the right, \(f(x)\) approaches \(3\).
\(\textbf{27)}\) \(\displaystyle \lim_{x\to1}f(x)=0\), Describe the limit in words. The answer is: As \(x\) approaches \(1\) from both sides, \(f(x)\) approaches \(0\). There is no \(+\) or \(–\) superscript, so x approaches 1 from both sides.
The function values approach \(0\).
As \(x\) approaches \(1\), \(f(x)\) approaches \(0\).
The function values approach \(0\).
As \(x\) approaches \(1\), \(f(x)\) approaches \(0\).
\(\textbf{28)}\) \(\displaystyle \lim_{x\to-2}f(x)=\infty\), Describe the limit in words. The answer is: As \(x\) approaches \(-2\) from both sides, the graph goes up without bound. There is no one-sided superscript, so x approaches -2 from both sides.
\(\infty\) means the graph goes up without bound.
As \(x\) approaches \(-2\), the graph goes up without bound.
\(\infty\) means the graph goes up without bound.
As \(x\) approaches \(-2\), the graph goes up without bound.
\(\textbf{29)}\) \(\displaystyle \lim_{x\to7}f(x)=-\infty\), Describe the limit in words. The answer is: As \(x\) approaches \(7\) from both sides, the graph goes down without bound. There is no one-sided superscript, so x approaches 7 from both sides.
\(-\infty\) means the graph goes down without bound.
As \(x\) approaches \(7\), the graph goes down without bound.
\(-\infty\) means the graph goes down without bound.
As \(x\) approaches \(7\), the graph goes down without bound.
\(\textbf{30)}\) \(\displaystyle \lim_{x\to\infty}f(x)=-\infty\), Describe the limit in words. The answer is: As the graph goes to the right, it goes down without bound. \(x\to\infty\) means far to the right.
\(f(x)\to-\infty\) means the graph goes down without bound.
As the graph goes to the right, it goes down without bound.
\(f(x)\to-\infty\) means the graph goes down without bound.
As the graph goes to the right, it goes down without bound.
\(\textbf{31)}\) \(\displaystyle \lim_{x\to-\infty}f(x)=\infty\), Describe the limit in words. The answer is: As the graph goes to the left, it goes up without bound. \(x\to-\infty\) means far to the left.
\(f(x)\to\infty\) means the graph goes up without bound.
As the graph goes to the left, it goes up without bound.
\(f(x)\to\infty\) means the graph goes up without bound.
As the graph goes to the left, it goes up without bound.
\(\textbf{32)}\) \(\displaystyle \lim_{x\to0^+}f(x)=4\), Describe the limit in words. The answer is: As \(x\) approaches \(0\) from the right, \(f(x)\) approaches \(4\). \(0^+\) means approach 0 from the right.
The function values approach \(4\).
As \(x\) approaches \(0\) from the right, \(f(x)\) approaches \(4\).
The function values approach \(4\).
As \(x\) approaches \(0\) from the right, \(f(x)\) approaches \(4\).
\(\textbf{33)}\) \(\displaystyle \lim_{x\to0^-}f(x)=-4\), Describe the limit in words. The answer is: As \(x\) approaches \(0\) from the left, \(f(x)\) approaches \(-4\). \(0^-\) means approach 0 from the left.
The function values approach \(-4\).
As \(x\) approaches \(0\) from the left, \(f(x)\) approaches \(-4\).
The function values approach \(-4\).
As \(x\) approaches \(0\) from the left, \(f(x)\) approaches \(-4\).
\(\textbf{34)}\) \(\displaystyle \lim_{x\to10^-}f(x)=1\), Describe the limit in words. The answer is: As \(x\) approaches \(10\) from the left, \(f(x)\) approaches \(1\). \(10^-\) means approach 10 from the left.
The function values approach \(1\).
As \(x\) approaches \(10\) from the left, \(f(x)\) approaches \(1\).
The function values approach \(1\).
As \(x\) approaches \(10\) from the left, \(f(x)\) approaches \(1\).
\(\textbf{35)}\) \(\displaystyle \lim_{x\to10^+}f(x)=1\), Describe the limit in words. The answer is: As \(x\) approaches \(10\) from the right, \(f(x)\) approaches \(1\). \(10^+\) means approach 10 from the right.
The function values approach \(1\).
As \(x\) approaches \(10\) from the right, \(f(x)\) approaches \(1\).
The function values approach \(1\).
As \(x\) approaches \(10\) from the right, \(f(x)\) approaches \(1\).
\(\textbf{36)}\) \(\displaystyle \lim_{x\to3}f(x)=9\), Describe the limit in words. The answer is: As \(x\) approaches \(3\) from both sides, \(f(x)\) approaches \(9\). There is no one-sided superscript, so x approaches 3 from both sides.
The function values approach \(9\).
As \(x\) approaches \(3\), \(f(x)\) approaches \(9\).
The function values approach \(9\).
As \(x\) approaches \(3\), \(f(x)\) approaches \(9\).
\(\textbf{37)}\) \(\displaystyle \lim_{x\to\infty}f(x)=-1\), Describe the limit in words. The answer is: As the graph goes to the right, it approaches \(y=-1\). \(x\to\infty\) means far to the right.
\(f(x)\to-1\) means the y-values approach -1.
As the graph goes to the right, it approaches \(y=-1\).
\(f(x)\to-1\) means the y-values approach -1.
As the graph goes to the right, it approaches \(y=-1\).
\(\textbf{38)}\) \(\displaystyle \lim_{x\to-\infty}f(x)=0\), Describe the limit in words. The answer is: As the graph goes to the left, it approaches the x-axis. \(x\to-\infty\) means far to the left.
\(f(x)\to0\) means the y-values approach 0.
As the graph goes to the left, it approaches the x-axis.
\(f(x)\to0\) means the y-values approach 0.
As the graph goes to the left, it approaches the x-axis.
\(\textbf{39)}\) \(\displaystyle \lim_{x\to2^+}f(x)=-\infty\), Describe the limit in words. The answer is: As \(x\) approaches \(2\) from the right, the graph goes down without bound. \(2^+\) means approach 2 from the right.
\(-\infty\) means the graph goes down without bound.
As \(x\) approaches \(2\) from the right, the graph goes down without bound.
\(-\infty\) means the graph goes down without bound.
As \(x\) approaches \(2\) from the right, the graph goes down without bound.
\(\textbf{40)}\) \(\displaystyle \lim_{x\to2^-}f(x)=\infty\), Describe the limit in words. The answer is: As \(x\) approaches \(2\) from the left, the graph goes up without bound. \(2^-\) means approach 2 from the left.
\(\infty\) means the graph goes up without bound.
As \(x\) approaches \(2\) from the left, the graph goes up without bound.
\(\infty\) means the graph goes up without bound.
As \(x\) approaches \(2\) from the left, the graph goes up without bound.
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\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Limits on Graphs}\)
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