A courier charges for a parcel by its mass w in grams (g)…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A courier charges for a parcel by its mass in grams (g). The price in rupees is for , then for , then for . A shopkeeper asks what happens to the price as the mass comes close to g. Find the left-hand limit and the right-hand limit of as g. Then find the price of a parcel of exactly g. Say whether the limit of at g exists.
Need a reminder? The key ideaShow
A limit looks at the values of near , never at itself. The limit exists when the left-hand and right-hand limits are equal. For a polynomial, the limit at is its value at .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Given. A price that jumps at g, g and g.
Plan. Read the band on each side of g.
Step 1. From the left. For just below g, the price is rupees. So the left-hand limit is rupees.
Step 2. From the right. For just above g, the price is rupees. The right-hand limit is rupees.
Step 3. The value. At exactly g the rule applies, so rupees.
Step 4. The limit. The two one-sided limits are and . They differ, so the limit at g does not exist.
Watch out. The price at g equals the left-hand limit, but that is not enough. The limit needs both sides to agree.
Answer: rupees from the left, rupees from the right, rupees at g, and no limit.
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