Let f(x)=(x^2-1)/(x-1), x≠ 1 3, x=1. Which of the following is true about the continuity of f at x=1?

f is continuous from left but not right at x=1

f is continuous from right but not left at x=1

Let $f (x) = {\frac{x ^{2} - 1}{x - 1}, 3, x \neq = 1 x = 1$ . Which of the following is true about the continuity of $f$ at $x = 1$ ?

easy MCQ 2 marks Class 12 · Mathematics · Continuity and Differentiability

Answer

f is continuous at x=1

The correct answer is f is continuous at x=1 — Since the limit does not equal the function value, $f$ is discontinuous at $x = 1$ .

First simplify the expression for $x \neq = 1$ : $\frac{x ^{2} - 1}{x - 1} = \frac{( x - 1 ) ( x + 1 )}{x - 1} = x + 1$ .
Compute the limit as $x \to 1$ of the simplified form: $lim_{x \to 1} (x + 1) = 2$ .
Evaluate the function value at $x = 1$ : $f (1) = 3$ .
Compare the limit and the function value: $lim_{x \to 1} f (x) = 2 \neq = f (1) = 3$ .
Since the limit does not equal the function value, $f$ is discontinuous at $x = 1$ .

Common mistake: Failing to cancel the factor

(x - 1)

and mistakenly concluding

f

is continuous

Concept tested: Continuity of piecewise functions