Q.A sales agent earns commission $g(x) = 500$ (a flat amount, in rupees) for sales $x \le 50{,}000$, and $g(x) = 0.02x$ for sales $x > 50{,}000$. Find the left-hand limit and the right-hand limit of $g(x)$ as $x \to 50{,}000$, and state whether the limit exists at that point.

Below the threshold the commission is a flat $500$; above it, the commission is $2\%$ of sales — these two rules do not meet at the threshold, so we check both one-sided limits.

LHL: $\lim_{x\to50000^-} 500 = 500$. RHL: $\lim_{x\to50000^+} 0.02x = 0.02(50000)=1000$. Since $500 \neq 1000$, the limit does not exist at $x=50{,}000$.

Step 1 — Left-hand limit. For $x\to50000^-$ (sales just below $\text{₹}50{,}000$), the flat-commission rule $g(x)=500$ applies for every such $x$ (it is a constant, so its limit is simply that constant):

Step 2 — Right-hand limit. For $x\to50000^+$ (sales just above $\text{₹}50{,}000$), the percentage rule $g(x)=0.02x$ applies:

Step 3 — Compare. LHL $=500$ and RHL $=1000$ are not equal, so by the existence rule (§2), the two-sided limit does not exist at $x=50{,}000$ — even though each one-sided limit individually is a perfectly ordinary, well-defined number.

Business reading: this is a genuine "jump" in the commission structure exactly at the ₹50,000 threshold — an agent who sells ₹49,999 earns ₹500 flat, while an agent who sells ₹50,001 earns roughly ₹1,000 (2% of sales) — a discontinuity a business might want to smooth out, which is precisely what the limit calculation has revealed.

Independent check: evaluate each piece at values very close to the threshold — e.g. $g(49{,}999)=500$ and $g(50{,}001)=0.02(50{,}001)=1000.02$ — confirming the two sides are heading toward genuinely different numbers ($500$ vs. approximately $1000$), not converging.

A frequent error is assuming a limit always exists as long as the function is defined at the point ($g(50{,}000)=500$ IS defined here, using the $x\le50{,}000$ piece) — but the EXISTENCE of $g(50{,}000)$ has nothing to do with whether the LIMIT exists; the limit depends only on the one-sided approach values, which here genuinely disagree.