Question 17 of 70beginner🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→0xsec⁡x\lim_{x \to 0} x \sec x

Correct Answer

0

Exercise: EXERCISE 12.1 | Q: 19 | (Page Page 22)
For More Understanding

Explanation

The question asks to evaluate the limit of x sec x as x approaches 0. Using the definition sec x = 1/cos x, we can rewrite the expression. From the context, we know that lim(x→0) cos x = 1. Using the property of limits shown in the context (where limits can be separated for sums and products), we can evaluate this limit by separating it into lim(x→0) x divided by lim(x→0) cos x. Since lim(x→0) x = 0 and lim(x→0) cos x = 1, the result is 0/1 = 0.

Solution Steps

  1. Step 1: Rewrite sec x as 1/cos x, so lim(x→0) x sec x = lim(x→0) x/cos x

  2. Step 2: Apply the limit property: lim(x→0) x/cos x = lim(x→0) x ÷ lim(x→0) cos x

  3. Step 3: Since lim(x→0) x = 0 and lim(x→0) cos x = 1 (from context)

  4. Step 4: Therefore, lim(x→0) x sec x = 0/1 = 0

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