Questions: lim as x approaches 0 of (sin^2(x) + sec(x))

Transcript text: (f) $\lim _{x \rightarrow 0}\left(\sin ^{2}(x)+\sec (x)\right)$

Solution

Solution Steps

To find the limit of the given expression as $ x $ approaches 0, we need to evaluate the behavior of each component of the expression separately. The $\sin^2(x)$ term approaches 0 as $ x $ approaches 0. The $\sec(x)$ term, which is $1/\cos(x)$, approaches 1 as $ x $ approaches 0 because $\cos(0) = 1$. Therefore, the limit of the entire expression is the sum of these limits.

Step 1: Evaluate $\sin^2(x)$ as $x \to 0$

As $x$ approaches 0, $\sin(x)$ approaches 0. Therefore, $\sin^2(x)$ also approaches 0.

Step 2: Evaluate $\sec(x)$ as $x \to 0$

The secant function is defined as $\sec(x) = \frac{1}{\cos(x)}$. As $x$ approaches 0, $\cos(x)$ approaches 1. Therefore, $\sec(x)$ approaches $\frac{1}{1} = 1$.

Step 3: Combine the Limits

The original expression is $\sin^2(x) + \sec(x)$. As $x$ approaches 0, the limit of $\sin^2(x)$ is 0 and the limit of $\sec(x)$ is 1. Therefore, the limit of the entire expression is: \[ \lim_{x \to 0} \left(\sin^2(x) + \sec(x)\right) = 0 + 1 = 1 \]

Final Answer

\[ \boxed{1} \]

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