Calculus Limits

Question

I am going crazy about this; can you please help and explain your answer

lim y→5  1/y-1/5

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y-5

Answer 1

This is an example of evaluating a limit.

Step 0: Problem

lim_y→₅ ((1/y – 1/5) / (y – 5))

Note that if we were to try to evaluate the limit by substituting y = 5, we would get 0 / 0. This is indeterminate. We must use algebra now.

Step 1: Make a Common Denominator in the Numerator

lim_y→₅ ((5/(5y) – y/(5y)) / (y – 5))

Step 2: Combine the Two Fractions in the Numerator

lim_y→₅ (((5 - y) / (5y)) / (y - 5))

Step 3: Factor Out a Negative One From the Numerator

lim_y→₅ ((-(y - 5) / (5y)) / (y - 5))

Step 4: Cancel the (y - 5) Terms

lim_y→₅ (-1 / 5y)

Step 5: Evaluate the Limit

Now that the (y - 5) terms have been canceled, we can finally evaluate the limit by substituting y = 5.

lim_y→₅ (-1 / 5y) = -1 / 25