LimitsJEE Main

Concept

A limit describes the value a function approaches as the input nears a point — independent of the value (if any) at that point. Here $f(x)=\dfrac{x^2-a^2}{x-a}$ is undefined at $x=a$ (it gives $\tfrac{0}{0}$), yet the limit exists and equals $2a$.

Key formula

Derivation

Factor the numerator as a difference of squares: $x^2-a^2=(x-a)(x+a)$.

For $x\neq a$ the $(x-a)$ cancels, leaving $f(x)=x+a$.

The limit only cares about $x$ near $a$ (not equal), so we may use the simplified form: $\lim_{x\to a}(x+a)=2a$. The graph is the line $y=x+a$ with a single hole at $(a,2a)$.

Scenarios to explore

• Limits — Approaching a value — epsilon-delta intuition.

Real-world applications

• Evaluating instantaneous rates (derivatives) which are limits of 0/0 form.
• Removable discontinuities in rational functions.
• Defining continuity: f is continuous at a iff the limit equals f(a).

JEE exam tips

• 0/0 is indeterminate, not zero or infinity — simplify, factor, or use L'Hôpital.
• A two-sided limit exists only if the left and right limits agree.

Common mistakes

• Substituting $x=a$ first and concluding the limit 'does not exist' because of 0/0.
• Confusing the limit (2a) with the function value at a (undefined).
• Cancelling $(x-a)$ without noting the restriction $x\neq a$.

Exam traps to avoid

• A hole (removable) vs a jump (non-removable) discontinuity — only the hole keeps the limit.
• $\lim_{x\to a}f(x)$ can exist even when $f(a)$ is undefined or different.