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A function f is said to be removable dis...

A function f is said to be removable discontinuity at x = 0, if `lim_(x to 0)f(x)` exists and

A

`lim_(x to 0) f(x)=f(a)`

B

`lim_(x to 0)f(x) ne f(a)`

C

`lim_(x to 0) f(x)= 0`

D

None of these

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The correct Answer is:
To determine if a function \( f \) has a removable discontinuity at \( x = 0 \), we need to follow these steps: ### Step 1: Understand the Definition A function \( f \) has a removable discontinuity at \( x = 0 \) if: 1. The limit \( \lim_{x \to 0} f(x) \) exists. 2. The value of the function at that point, \( f(0) \), is not equal to the limit, i.e., \( f(0) \neq \lim_{x \to 0} f(x) \). ### Step 2: Check the Limit To check if the limit exists, we need to evaluate: \[ \lim_{x \to 0} f(x) \] If this limit exists (is a finite number), we can proceed to the next step. ### Step 3: Evaluate the Function at the Point Next, we need to find the value of the function at \( x = 0 \): \[ f(0) \] ### Step 4: Compare the Limit and the Function Value Finally, we compare the limit and the function value: - If \( f(0) \neq \lim_{x \to 0} f(x) \), then the function has a removable discontinuity at \( x = 0 \). ### Conclusion If both conditions are satisfied, we conclude that the function has a removable discontinuity at \( x = 0 \). ---
To determine if a function \( f \) has a removable discontinuity at \( x = 0 \), we need to follow these steps: ### Step 1: Understand the Definition A function \( f \) has a removable discontinuity at \( x = 0 \) if: 1. The limit \( \lim_{x \to 0} f(x) \) exists. 2. The value of the function at that point, \( f(0) \), is not equal to the limit, i.e., \( f(0) \neq \lim_{x \to 0} f(x) \). ### Step 2: Check the Limit ...
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