Find Limit of f(x) as x Approaches 0: Solutions Explained

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In summary, the limit of f(x) when x approaches 0 is 0.25. The given function is (1/x^3){(1+tanx)^0.5 - (1+sinx)^0.5}. Some attempted solutions include using the conjugate and substitution, but they did not work. By replacing sin(x) and tan(x) with their first terms in their Taylor series around 0, the limit can be simplified to 0.25.
  • #1
lkh1986
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Find limit of f(x) when x approaches 0.

Given that f(x) is (1/x^3){(1+tanx)^0.5 - (1+sinx)^0.5}

The given answer is 0.25, but can somebody show me the solutions? I try the conjugate, and nothing works. Then I try to substitute t=(1+tanx)^0.5 and others, can't work, too.
 
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  • #2
If you multiply numerator and denominator with the complement expression, you get after simplifying:

[tex]\frac{{\frac{{\tan x - \sin x}}{{x^3 }}}}{{\sqrt {1 + \tan x} + \sqrt {1 + \sin x} }}[/tex]

Replacing sin(x) and tan(x) by the first terms of their Taylor series arround 0 so that the difference isn't 0, is x³/2. So you get:

[tex]
\mathop {\lim }\limits_{x \to 0} \frac{{\frac{{\tan x - \sin x}}{{x^3 }}}}{{\sqrt {1 + \tan x} + \sqrt {1 + \sin x} }} = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{1}{2}}}{{\sqrt {1 + \tan x} + \sqrt {1 + \sin x} }}
[/tex]
 
  • #3
lkh1986 said:
Find limit of f(x) when x approaches 0.

Given that f(x) is (1/x^3){(1+tanx)^0.5 - (1+sinx)^0.5}
Wait a minute, what is your f(x)?
Is it
[tex]\frac {\sqrt{1+tanx} - \sqrt{1+sinx}}{x^3}[/tex]?
 

What is the definition of a limit?

The limit of a function f(x) as x approaches a is the value that f(x) gets closer and closer to as x gets closer and closer to a. It is denoted by limx→a f(x).

How do you find the limit of a function as x approaches a specific value?

To find the limit of a function f(x) as x approaches a, you can use the following steps:

  • Substitute the value of a into the function f(x) to get f(a).
  • Simplify the resulting expression as much as possible.
  • If the resulting expression is undefined, try approaching a from both the left and right sides to see if the limit exists.
  • If the resulting expression is a finite number, then that is the limit of f(x) as x approaches a.
  • If the resulting expression is an infinite value, then the limit does not exist.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit is the limit of a function at a specific point from only one direction, either the left or the right side. A two-sided limit, also known as a two-sided limit, is the limit of a function at a specific point from both the left and the right sides. One-sided limits are used when the function is not defined on both sides of the point, while two-sided limits are used when the function is defined on both sides of the point.

What are the algebraic rules for finding limits?

Some of the algebraic rules for finding limits include:

  • The limit of a constant is the constant itself.
  • The limit of a sum or difference of two functions is the sum or difference of the limits of the individual functions.
  • The limit of a product of two functions is the product of the limits of the individual functions.
  • The limit of a quotient of two functions is the quotient of the limits of the individual functions, as long as the limit of the denominator is not 0.
  • The limit of a power of a function is the power of the limit of the function.
  • The limit of a composite function is the composite of the limits of the individual functions.

How can you use graphs to determine the limit of a function?

You can use the graph of a function to determine the limit at a specific point by looking at the behavior of the function as it approaches that point. If the graph has a "hole" or a point where the function is not defined, you can approach the point from both sides to see if the limit exists. If the graph has a vertical asymptote, the limit does not exist. If the graph approaches a specific value as x approaches the point from both sides, then that value is the limit. If the graph oscillates or has no clear behavior, the limit does not exist.

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