Evaluate the Limit ( limit as x approaches 0 of sec(x)-1)/x

Problem

(lim_x→0)((sec(x)−1)/x)

Solution

1. Identify the form of the limit by substituting x=0 into the expression.

sec(0)−1=1−1=0

2. Recognize that the expression results in the indeterminate form 0/0 allowing the use of L'Hôpital's Rule.

(lim_x→0)((sec(x)−1)/x)=(lim_x→0)((d(sec(x))−1)/d(x)/d(x)/d(x))

3. Differentiate the numerator and the denominator separately.

(d(sec(x))−1)/d(x)=sec(x)*tan(x)

d(x)/d(x)=1

4. Rewrite the limit using the derivatives found in the previous step.

(lim_x→0)((sec(x)*tan(x))/1)

5. Evaluate the limit by substituting x=0 into the simplified expression.

sec(0)*tan(0)=1⋅0=0

Final Answer

(lim_x→0)((sec(x)−1)/x)=0

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