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Suppose that a function
f(x) is defined for all real values of
x except
x=c. Can anything be said about the existence of
lim_(x rarr c)f(x) ? Give reasons.

Suppose that a function $f(x)$ is defined for all real values of $x$ except $x=c$ . Can anything be said about the existence of $\lim_{x \rightarrow c}f(x)$ ? Give reasons.

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Find the magnitude of a three-dimensional vector

Full solution

Q. Suppose that a function

f(x)

is defined for all real values of

x

except

x=c

. Can anything be said about the existence of

\lim_{x \rightarrow c}f(x)

? Give reasons.

Function Behavior at $x=c$ : Just because $f(x)$ is undefined at $x=c$ doesn't mean the limit as $x$ approaches $c$ doesn't exist. The limit could still exist if the function approaches a particular value from both sides as $x$ gets closer to $c$ .
Example with $f(x)=x^2-1/x-1$ : For example, consider the function $f(x) = \frac{x^2 - 1}{x - 1}$ . This function is undefined at $x=1$ , but if you simplify it, you get $f(x) = x + 1$ , which is defined at $x=1$ . So the limit as $x$ approaches $1$ exists and is $2$ .
Different Behavior on Left and Right: However, if the function $f(x)$ behaves differently from the left side of $c$ compared to the right side, or if it goes to infinity, then the limit as $x$ approaches $c$ does not exist.
Analyzing $f(x)$ Behavior: To determine the existence of the limit, we would need to analyze the behavior of $f(x)$ as $x$ approaches $c$ from both sides. Without more information about $f(x)$ , we can't say for sure whether the limit exists.

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