Evaluate the Limit

Problem

(lim_y→1)(sec(y*sec2(y)−tan2(y)−1))

Solution

1. Identify the expression inside the secant function and recall the fundamental trigonometric identity relating secant and tangent.

sec2(y)−tan2(y)=1

2. Substitute the identity into the expression to simplify the argument of the limit.

y*sec2(y)−tan2(y)−1=y*sec2(y)−(tan2(y)+1)

3. Apply the identity tan2(y)+1=sec2(y) to further reduce the expression.

y*sec2(y)−sec2(y)

4. Factor out the common term sec2(y) from the expression.

(y−1)*sec2(y)

5. Evaluate the limit by substituting y=1 into the simplified expression, noting that the secant function is continuous at y=1

(lim_y→1)(y−1)*sec2(y)=(1−1)*sec2(1)

6. Calculate the resulting value of the argument.

0⋅sec2(1)=0

7. Compute the final value of the secant function at the evaluated argument.

sec(0)=1

Final Answer

(lim_y→1)(sec(y*sec2(y)−tan2(y)−1))=1

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