Question:
Examine the continuity of the function f(x) = 2x2-1 at x = 3.
Examine the continuity of the function f(x) = 2x2-1 at x = 3.
Updated On: Jun 26, 2024
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Approach Solution - 1
The given function is f(x) = 2x2-1
At x = 3, f(x) = f(3) = 2x32-1 = 17
\(\lim\limits_{x \to 3}\) f(x) = \(\lim\limits_{x \to 3}\) (2x2-1) = 2x32-1 = 17
∴\(\lim\limits_{x \to 3}\) f(x) = f(3)
Therefore, f is continuous at x=3.
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Approach Solution -2
f(x) is continuous at x = 3 if
\(\lim\limits_{x\rightarrow3}f(x)=f(3)\)
As L.H.S = R.H.S
Hence, f is continuous at x = 3.
\(\lim\limits_{x\rightarrow3}f(x)=f(3)\)
| L.H.S | R.H.S |
| \(\lim\limits_{x\rightarrow3}f(x)\) \(=\lim\limits_{x\rightarrow3}(2x^2-1)\) Putting x = 3 = 2(3)2 - 1 = 2 × 9 - 1 = 17 | f(3) = 2(3)2 - 1 = 2 × 9 - 1 =17 |
Hence, f is continuous at x = 3.
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.