Answer :
To find the limit \(\lim_{x \to 0} \frac{x^2}{\sec x - 1}\), we'll go through the following steps:
1. Recall the definition of \(\sec(x)\):
\(\sec(x) = \frac{1}{\cos(x)}\).
2. Analyze the behavior of \(\sec(x)\) as \(x \to 0\):
As \(x\) approaches 0, \(\cos(x)\) approaches 1. Therefore, \(\sec(x) = \frac{1}{\cos(x)}\) approaches 1 as well.
3. Rewrite the limit:
Substitute \(\sec(x)\) with \(\frac{1}{\cos(x)}\):
[tex]\[ \lim_{x \to 0} \frac{x^2}{\sec(x) - 1} = \lim_{x \to 0} \frac{x^2}{\frac{1}{\cos(x)} - 1} \][/tex]
4. Simplify the expression:
Get a common denominator for the expression in the denominator:
[tex]\[ \frac{1}{\cos(x)} - 1 = \frac{1 - \cos(x)}{\cos(x)} \][/tex]
Thus, the limit becomes:
[tex]\[ \lim_{x \to 0} \frac{x^2}{\frac{1 - \cos(x)}{\cos(x)}} = \lim_{x \to 0} \frac{x^2 \cos(x)}{1 - \cos(x)} \][/tex]
5. Use the small-angle approximation for \(\cos(x)\):
For small values of \(x\), \(\cos(x)\) can be approximated by \(1 - \frac{x^2}{2}\):
[tex]\[ 1 - \cos(x) \approx 1 - \left(1 - \frac{x^2}{2}\right) = \frac{x^2}{2} \][/tex]
6. Substitute the approximation into the limit:
[tex]\[ \lim_{x \to 0} \frac{x^2 \cos(x)}{\frac{x^2}{2}} = \lim_{x \to 0} \frac{x^2 \cos(x)}{\frac{x^2}{2}} = \lim_{x \to 0} \frac{2 x^2 \cos(x)}{x^2} \][/tex]
7. Simplify the limit:
[tex]\[ \lim_{x \to 0} 2 \cos(x) = 2 \cdot \lim_{x \to 0} \cos(x) \][/tex]
8. Evaluate the limit:
As \(x \to 0\), \(\cos(x) \to 1\). Therefore:
[tex]\[ 2 \cdot 1 = 2 \][/tex]
Thus, the limit is:
[tex]\[ \lim_{x \to 0} \frac{x^2}{\sec(x) - 1} = 2 \][/tex]
1. Recall the definition of \(\sec(x)\):
\(\sec(x) = \frac{1}{\cos(x)}\).
2. Analyze the behavior of \(\sec(x)\) as \(x \to 0\):
As \(x\) approaches 0, \(\cos(x)\) approaches 1. Therefore, \(\sec(x) = \frac{1}{\cos(x)}\) approaches 1 as well.
3. Rewrite the limit:
Substitute \(\sec(x)\) with \(\frac{1}{\cos(x)}\):
[tex]\[ \lim_{x \to 0} \frac{x^2}{\sec(x) - 1} = \lim_{x \to 0} \frac{x^2}{\frac{1}{\cos(x)} - 1} \][/tex]
4. Simplify the expression:
Get a common denominator for the expression in the denominator:
[tex]\[ \frac{1}{\cos(x)} - 1 = \frac{1 - \cos(x)}{\cos(x)} \][/tex]
Thus, the limit becomes:
[tex]\[ \lim_{x \to 0} \frac{x^2}{\frac{1 - \cos(x)}{\cos(x)}} = \lim_{x \to 0} \frac{x^2 \cos(x)}{1 - \cos(x)} \][/tex]
5. Use the small-angle approximation for \(\cos(x)\):
For small values of \(x\), \(\cos(x)\) can be approximated by \(1 - \frac{x^2}{2}\):
[tex]\[ 1 - \cos(x) \approx 1 - \left(1 - \frac{x^2}{2}\right) = \frac{x^2}{2} \][/tex]
6. Substitute the approximation into the limit:
[tex]\[ \lim_{x \to 0} \frac{x^2 \cos(x)}{\frac{x^2}{2}} = \lim_{x \to 0} \frac{x^2 \cos(x)}{\frac{x^2}{2}} = \lim_{x \to 0} \frac{2 x^2 \cos(x)}{x^2} \][/tex]
7. Simplify the limit:
[tex]\[ \lim_{x \to 0} 2 \cos(x) = 2 \cdot \lim_{x \to 0} \cos(x) \][/tex]
8. Evaluate the limit:
As \(x \to 0\), \(\cos(x) \to 1\). Therefore:
[tex]\[ 2 \cdot 1 = 2 \][/tex]
Thus, the limit is:
[tex]\[ \lim_{x \to 0} \frac{x^2}{\sec(x) - 1} = 2 \][/tex]
