Calculating Limits Using Properties

• nycmathguy
In summary, calculating limits using properties involves using various mathematical rules and concepts to determine the behavior of a function at a specific point. These properties include the sum, difference, product, and quotient rules, as well as the power, exponential, and logarithmic properties. By applying these properties, one can simplify complex expressions and ultimately find the limit of a function as it approaches a certain value. This method is commonly used in calculus and is essential for understanding the behavior of functions and their graphs.
nycmathguy
Homework Statement
Use the properties of limits to find the limits.
Relevant Equations
N/A
Use properties of limits to find the limit.

lim (-3x + 1)^2
x→0

[lim (-3x + 1) as x→0 ]^2

[-3•lim(x) as x→0 + lim (1) as x→0]^2

[-3•0 + 1]^2

[0 + 1]^2

[1]^2 = 1

The limit is 1.

Looks good to me.

It would be easier to read these if you used latex. It's not that hard to learn and use, if you plan on posting for a while it's worth figuring it out.

SammyS
Office_Shredder said:
It would be easier to read these if you used latex. It's not that hard to learn and use, if you plan on posting for a while it's worth figuring it out.
Amen to that!

nycmathguy said:
Homework Statement:: Use the properties of limits to find the limits.
Relevant Equations:: N/A

lim (-3x + 1)^2
x→0
In LaTeX, this looks like ##\lim_{x \to 0}(-3x + 1)^2##
In rendered form it is ##\lim_{x \to 0}(-3x + 1)^2##

nycmathguy said:
Homework Statement:: Use the properties of limits to find the limits.
Relevant Equations:: N/A

Use properties of limits to find the limit.

lim (-3x + 1)^2
x→0

[lim (-3x + 1) as x→0 ]^2

[-3•lim(x) as x→0 + lim (1) as x→0]^2

[-3•0 + 1]^2

[0 + 1]^2

[1]^2 = 1

The limit is 1.
Technically, it's better the other way round:
$$\lim_{x \rightarrow 0} x = 0$$ $$\lim_{x \rightarrow 0} 3x = 3\lim_{x \rightarrow 0} x = 0$$ $$\lim_{x \rightarrow 0} (3x + 1) = \lim_{x \rightarrow 0} 3x +1 = 1$$ $$\lim_{x \rightarrow 0} (3x + 1)^2 = [\lim_{x \rightarrow 0} (3x +1)]^2 = 1^2 = 1$$
Note that the existence and calculation for each limit follows from the previous limit.

nycmathguy
PeroK said:
Technically, it's better the other way round:
$$\lim_{x \rightarrow 0} x = 0$$ $$\lim_{x \rightarrow 0} 3x = 3\lim_{x \rightarrow 0} x = 0$$ $$\lim_{x \rightarrow 0} (3x + 1) = \lim_{x \rightarrow 0} 3x +1 = 1$$ $$\lim_{x \rightarrow 0} (3x + 1)^2 = [\lim_{x \rightarrow 0} (3x +1)]^2 = 1^2 = 1$$
Note that the existence and calculation for each limit follows from the previous limit.
Look great.

What is the definition of a limit?

A limit is the value that a function approaches as the input approaches a specific value. It represents the behavior of the function near that specific input.

What are the properties used to calculate limits?

The properties used to calculate limits are the sum and difference properties, product property, quotient property, power property, and constant multiple property.

How do you use the sum and difference properties to calculate a limit?

The sum and difference properties state that the limit of a sum or difference of two functions is equal to the sum or difference of the limits of the individual functions. In other words, you can calculate the limit of a sum or difference by taking the limit of each function separately and then adding or subtracting the results.

What is the power property and how is it used to calculate limits?

The power property states that the limit of a power of a function is equal to the power of the limit of the function. In other words, to calculate the limit of a function raised to a power, you can take the limit of the function and raise it to the same power.

What is the constant multiple property and how is it used to calculate limits?

The constant multiple property states that the limit of a constant times a function is equal to the constant times the limit of the function. In other words, to calculate the limit of a function multiplied by a constant, you can take the limit of the function and then multiply it by the constant.

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