Limits and ContinuityPYQ June 24Question 2008 of 32
All Questions

The lim⁡x→2x2−4x+4x−2=\displaystyle \lim_{x \to 2} \frac{x^2-4x+4}{x-2} =

Options

A0
B1
C2
D0.5
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Correct Answer

✅ Option a — 0

All Options:

  • A0
  • B1
  • C2
  • D0.5

Detailed Solution & Explanation

To find the limit lim⁡x→2x2−4x+4x−2\displaystyle \lim_{x \to 2} \frac{x^2-4x+4}{x-2}:
We first simplify the algebraic expression in the numerator.
Notice that the numerator x2−4x+4\displaystyle x^2 - 4x + 4 is a perfect square trinomial:
x2−4x+4=(x−2)2x^2 - 4x + 4 = (x - 2)^2
Substituting this back into the limit expression:
lim⁡x→2(x−2)2x−2\lim_{x \to 2} \frac{(x-2)^2}{x-2}
For x≠2\displaystyle x \neq 2, we can cancel one factor of (x−2)\displaystyle (x - 2) from the numerator and denominator:
lim⁡x→2(x−2)\lim_{x \to 2} (x - 2)
Now we substitute x=2\displaystyle x = 2 directly into the simplified expression:
2−2=02 - 2 = 0
Thus, the limit evaluates to 0\displaystyle 0.
Therefore, the correct choice is **Option A**.

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