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In mathematics, a limit is a value that a function approaches as the input approaches some value. Limits are used to define derivatives, integrals, and other important concepts in calculus. In order for a limit to exist, the function must approach the same value from both the left and the right side of the input value. If the function approaches different values from the left and right sides, then the limit does not exist.
Find All Values Of C Such That The Limit Exists
In this article, we will find all values of c such that the limit of the function
$$f(x) = \begincases x^2 + c, & x \leq 0 \\ x – c, & x > 0 \endcases$$
exists.
The Limit of a Function
The limit of a function is a value that the function approaches as the input approaches some value. The limit is written as
$$\lim_x \to a f(x) = L$$
where x is the input, a is the value that x is approaching, and L is the limit. The limit exists if the left-hand limit and the right-hand limit are equal.
The left-hand limit is the limit of the function as x approaches a from the left side. It is written as
$$\lim_x \to a^- f(x) = L$$
The right-hand limit is the limit of the function as x approaches a from the right side. It is written as
$$\lim_x \to a^+ f(x) = L$$
If the left-hand limit and the right-hand limit are equal, then the limit exists and is equal to L.
The Limit of the Function f(x)
In order to find the values of c such that the limit of the function f(x) exists, we need to find the left-hand limit and the right-hand limit of the function.
The left-hand limit is
$$\limx \to 0^- f(x) = \limx \to 0^- (x^2 + c) = 0 + c = c$$
The right-hand limit is
$$\limx \to 0^+ f(x) = \limx \to 0^+ (x – c) = 0 – c = -c$$
The left-hand limit and the right-hand limit are not equal, so the limit of the function does not exist.
Conclusion
The limit of the function f(x) does not exist for any value of c. This is because the left-hand limit and the right-hand limit of the function are not equal.
FAQ
Q: What is a limit in mathematics?
A: A limit is a value that a function approaches as the input approaches some value.
Q: What is the left-hand limit of a function?
A: The left-hand limit is the limit of the function as the input approaches a value from the left side.
Q: What is the right-hand limit of a function?
A: The right-hand limit is the limit of the function as the input approaches a value from the right side.
Q: What does it mean for a limit to exist?
A: A limit exists if the left-hand limit and the right-hand limit of the function are equal.
Find All Values Of C Such That The Limit Exists
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