Limits

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Math › Limits

Questions 1 - 10

Considering the following piecewise function, what is $\lim_{x \rightarrow 4^+} y$ ,

$y = \begin{Bmatrix}x^2 & x < 4 \ x-1 & x = 4 \x+7 & x \geq 4\end{Bmatrix}$

Does not exist

Explanation

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

Solution:

In this case, we want to see the limit at , from the right. The limit exists, and the value corresponds with the function

Considering the following piecewise function, what is $\lim_{x \rightarrow 4^+} y$ ,

$y = \begin{Bmatrix}x^2 & x < 4 \ x-1 & x = 4 \x+7 & x \geq 4\end{Bmatrix}$

Does not exist

Explanation

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

Solution:

In this case, we want to see the limit at , from the right. The limit exists, and the value corresponds with the function

Considering the following piecewise function, what is $\lim_{x \rightarrow 4^+} y$ ,

$y = \begin{Bmatrix}x^2 & x < 4 \ x-1 & x = 4 \x+7 & x \geq 4\end{Bmatrix}$

Does not exist

Explanation

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

Solution:

In this case, we want to see the limit at , from the right. The limit exists, and the value corresponds with the function

Considering the following piecewise function, what is $\lim_{x \rightarrow 2^+} y,$ $y= \begin{Bmatrix}3 & x < 2 \ -2 & x \geq 2\end{Bmatrix}$

Does not exist

Explanation

In general, when you are looking for $\lim_{x \rightarrow a^+} y$ 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 , from the right. The limit exists, and the value is

Considering the following piecewise function, what is $\lim_{x \rightarrow 2^+} y,$ $y= \begin{Bmatrix}3 & x < 2 \ -2 & x \geq 2\end{Bmatrix}$

Does not exist

Explanation

In general, when you are looking for $\lim_{x \rightarrow a^+} y$ 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 , from the right. The limit exists, and the value is

Considering the following piecewise function, what is $\lim_{x \rightarrow 2^+} y,$ $y= \begin{Bmatrix}3 & x < 2 \ -2 & x \geq 2\end{Bmatrix}$

Does not exist

Explanation

In general, when you are looking for $\lim_{x \rightarrow a^+} y$ 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 , from the right. The limit exists, and the value is

Considering the following piecewise function, what is $\lim_{x \rightarrow 2^+} y,$ $y= \begin{Bmatrix}3 & x < 2 \ -2 & x \geq 2\end{Bmatrix}$

Does not exist

Explanation

In general, when you are looking for $\lim_{x \rightarrow a^+} y$ 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 , from the right. The limit exists, and the value is

$\begin{align*}&\text{Evaluate the following limit:}\&lim_{x\rightarrow1-}\frac{41\cdot (x - 1)^{2}}{46ln(x^{8})}\end{align*}$

$\infty$

$-\infty$

$\frac{41}{135424}$

$-\frac{41}{135424}$

Explanation

$\begin{align*}&\text{Examining this problem, we find that both}\&\text{numerator and denominator have zero values at the limit specified.}\&\text{Noting this, L'Hopital's method may be of use:}\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\&\frac{41\cdot (x - 1)^{2}}{46ln(x^{8})}\rightarrow\frac{0}{0}\&\frac{82x - 82}{\frac{368}{x}}\rightarrow\frac{0}{368}=0\&lim_{x\rightarrow1-}\frac{41\cdot (x - 1)^{2}}{46ln(x^{8})}=0\end{align*}$