Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #281 : Limits

Considering the following piecewise function, what is \(\displaystyle \lim_{x \rightarrow 11^+} y\),\(\displaystyle y = \begin{Bmatrix}8 & x < 11 \\ 2 & x = 11 \\-6 & x \geq 11\end{Bmatrix}\).

Possible Answers:

\(\displaystyle -6\)

\(\displaystyle 2\)

\(\displaystyle 8\)

Does not exist

Correct answer:

\(\displaystyle -6\)

Explanation:

In general, when you are looking for \(\displaystyle \lim_{x \rightarrow a^+} y\) you are looking to see whether the limit of y exists to the right, and if it does, what is the value.

Solution:

In this case, we want to see the limit at \(\displaystyle x=11\), from the right. The limit exists, and it corresponds to the function \(\displaystyle -6\). Since this function has no inputs, the limit is \(\displaystyle -6\).

Example Question #281 : Calculus Ii

What is the \(\displaystyle \lim_{x \rightarrow 0^-} y\), for \(\displaystyle y=\frac{x^2+2}{x-2}\).

Possible Answers:

\(\displaystyle 1\)

Does not exist

\(\displaystyle 0\)

\(\displaystyle -1\)

Correct answer:

\(\displaystyle -1\)

Explanation:

As you go to \(\displaystyle x=0\) from the left, what value do you get closer to?

The limit of \(\displaystyle x=0\) from the left is 

\(\displaystyle \frac{0^2 + 2}{0-2} = -1\).

Example Question #283 : Limits

Evaluate the following limit:

\(\displaystyle \lim_{x\rightarrow \pi^{-}}f(x), f(x)=\left\{\begin{matrix} \cos(x), x\leq \pi\\ x, x> \pi \end{matrix}\right.\)

Possible Answers:

\(\displaystyle -\pi\)

\(\displaystyle 1\)

\(\displaystyle -1\)

\(\displaystyle \pi\)

The limit does not exist

Correct answer:

\(\displaystyle -1\)

Explanation:

To evaluate the limit, we must first see whether the limit is right or left sided. The negative sign "exponent" on \(\displaystyle \pi\) indicates that we are evaluating the limit from the left side, using \(\displaystyle x\) values slightly less than \(\displaystyle \pi\). So, the part of the piecewise function we will use is the first one; when we evaluate the limit, we get an answer of \(\displaystyle -1\) 

(\(\displaystyle \cos(\pi)=-1\)).

Example Question #284 : Limits

Considering the following piecewise function, what is \(\displaystyle \lim_{x \rightarrow 9^+} y\),

\(\displaystyle y = \begin{Bmatrix}x^2 & x < 9 \\ x+20 & x = 9 \\x^2+9 & x \geq 9\end{Bmatrix}\)

Possible Answers:

\(\displaystyle 29\)

\(\displaystyle 90\)

Does not exist

\(\displaystyle 81\)

Correct answer:

\(\displaystyle 90\)

Explanation:

In general, when you are looking for \(\displaystyle \lim_{x \rightarrow a^+} y\) you are looking to see whether the limit of y exists to the right, and if it does, what is the value.

Solution:

In this case, we want to see the limit at \(\displaystyle x=9\) , from the right. The limit exists, and the value corresponds to the function \(\displaystyle x^2+9\)

\(\displaystyle x^2+9 = 9^2 + 9 = 81 + 9 = 90\)

Example Question #285 : Limits

Considering the following piecewise function, what is \(\displaystyle \lim_{x \rightarrow 7^-} y\),

\(\displaystyle y = \begin{Bmatrix}x-5 & x < 7 \\ x^2+1 & x = 7 \\x+12 & x \geq 7\end{Bmatrix}\)

Possible Answers:

\(\displaystyle 19\)

\(\displaystyle 50\)

\(\displaystyle 2\)

Does not exist

Correct answer:

\(\displaystyle 2\)

Explanation:

In general, when you are looking for \(\displaystyle \lim_{x \rightarrow a^-} y\) you are looking to see whether the limit of y exists to the left, and if it does, what is the value.

Solution:

In this case, we want to see the limit at \(\displaystyle x=7\), from the left. The limit exists, and the value corresponds to the function \(\displaystyle x-5\)

\(\displaystyle x-5 = 7-5 = 2\)

Example Question #286 : Limits

Considering the following piecewise function, what is \(\displaystyle \lim_{x \rightarrow 3^-} y\),

\(\displaystyle y = \begin{Bmatrix}x-4 & x < 3 \\ x+7 & x = 3 \\x^3 & x \geq 3\end{Bmatrix}\)

Possible Answers:

\(\displaystyle 10\)

\(\displaystyle 27\)

\(\displaystyle -1\)

Does not exist

Correct answer:

\(\displaystyle -1\)

Explanation:

In general, when you are looking for \(\displaystyle \lim_{x \rightarrow a^-} y\) you are looking to see whether the limit of y exists to the left, and if it does, what is the value.

Solution:

In this case, we want to see the limit at \(\displaystyle x=3\), from the left. The limit exists, and the value correponds to the function \(\displaystyle x-4\)

\(\displaystyle x-4 = 3-4 = -1\)

Example Question #287 : Limits

Evaluate the following limit:

\(\displaystyle \lim_{x\rightarrow 7^+}f(x), f(x)=\left\{\begin{matrix} x+5, x\leq 7\\ \ln(x-7), x> 7 \end{matrix}\right.\)

Possible Answers:

The limit does not exist

\(\displaystyle -\infty\)

\(\displaystyle 12\)

\(\displaystyle 0\)

\(\displaystyle \infty\)

Correct answer:

\(\displaystyle -\infty\)

Explanation:

To evaluate the limit, we must first determine whether the limit is right or left sided. The positive sign "exponent" on 7 indicates that we are evaluating the limit from the right side, or using numbers slightly larger than 7. The part of the piecewise function corresponding to these values is the second function; when we evaluate the limit using that function, we approach \(\displaystyle -\infty\) (the natural log function approaches negative infinity as x approaches zero).

Example Question #288 : Limits

Find the limit if it exists

\(\displaystyle \lim_{x\rightarrow 2}f(x)\)

given the function

\(\displaystyle f(x)=\left\{\begin{matrix} x-2& x< 2 \,\,or\,\, x>2\\ 3&x=2 \end{matrix}\right.\)

Possible Answers:

\(\displaystyle 3\)

\(\displaystyle 0\)

\(\displaystyle DNE\)

Correct answer:

\(\displaystyle 0\)

Explanation:

The limit exists if 

 \(\displaystyle \lim_{x\rightarrow a^-}f(x) = \lim_{x\rightarrow a^+}f(x)\)

because 

 \(\displaystyle f(1.9)=1.9-2=-0.1\)

 \(\displaystyle f(1.99)=1.99-2=-0.01\)

 \(\displaystyle f(1.999)=1.999-2=-0.001\)

we see that 

 \(\displaystyle \lim_{x\rightarrow 2^-}f(x) = 0\)

because 

\(\displaystyle f(2.1)=2.1-2=0.1\) 

 \(\displaystyle f(2.01)=2.01-2=0.01\)

 \(\displaystyle f(2.001)=2.001-2=0.001\)

we see that 

\(\displaystyle \lim_{x\rightarrow 2^+}f(x) = 0\) 

since 

\(\displaystyle \lim_{x\rightarrow a^-}f(x) = \lim_{x\rightarrow a^+}f(x)=0\) 

we conclude that the limit exists and 

\(\displaystyle \lim_{x\rightarrow 2}f(x) = 0\)

Example Question #282 : Calculus Ii

Evaluate the following limit:

\(\displaystyle \lim_{x\rightarrow 2}f(x), f(x)=\left\{\begin{matrix} x^2+3, x< 2\\ 7\cos(x-2), x\geq 2 \end{matrix}\right.\)

Possible Answers:

\(\displaystyle \infty\)

\(\displaystyle 0\)

The limit does not exist

\(\displaystyle 7\)

Correct answer:

\(\displaystyle 7\)

Explanation:

To evaluate the limit, we must make sure that the same value is being approached from both sides. When we evaluate the limit from the left (the first part of the piecewise function) and the right (the second part of the piecewise function), we get the same value (\(\displaystyle 7\)) so the limit is equal to \(\displaystyle 7\).

Example Question #290 : Limits

Evaluate the following limit:

\(\displaystyle \lim_{x\rightarrow 2^+}f(x), f(x)=\left\{\begin{matrix}\sec(2-x), x< 2\\ \ln(x), x\geq 2 \end{matrix}\right.\)

Possible Answers:

\(\displaystyle -1\)

\(\displaystyle \ln(2)\)

The limit does not exist

\(\displaystyle 0\)

\(\displaystyle 1\)

Correct answer:

\(\displaystyle \ln(2)\)

Explanation:

To evaluate the limit, we must first determine whether the limit is right or left sided; the positive sign "exponent" on the \(\displaystyle 2\) indicates that the limit is right sided, or that we are approaching \(\displaystyle 2\) with values slightly greater than \(\displaystyle 2\). This corresponds to the second half of the piecewise function, and when we evaluate the limit using that function we get \(\displaystyle \ln(2)\)

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