Let a be an integer such that \(lim _{x→7} \frac{18−[1−x]}{[x−3a]}\) exists, where [t] is greatest integer \(≤ t\). Then \(a\) is equal to :
- -6
- -2
- 2
- 6
The Correct Option is A
Solution and Explanation
\(lim _{x→7} \frac{18−[1−x]}{[x−3a]}\)
exist & \(a∈l\).
\(lim_{x→7} \frac{17−[−x]}{[x−3a]}\)
exist
\(RHL\) = \(lim_{x→7} \frac{17−[−x]}{[x−3a]}\) = \(\frac{25}{7-3a} [a ≠\frac{7}{3}]\)
\(LHL\) = \(lim_{x→7-} \frac{17−[−x]}{[x−3a]}\)
= \(\frac{24}{6-3a} [ a≠2]\)
For limit to exist
\(LHL = RHL\)
\(\frac{25}{7−3a}=\frac{24}{6−3a}\)
\(⇒\frac{25}{7−3a}=\frac{8}{2−a}\)
\(∴ a = -6\)
Hence, the correct option is (A): \(-6\)
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Concepts Used:
Limits And Derivatives
Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
Properties of Derivatives:
Read More: Limits and Derivatives