Calculus Examples

$lim_{x \to 0} ((e^{x} - x - 1) \cos^{2} (\frac{2}{x})) + x + 1$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} (e^{x} - x - 1) \cos^{2} (\frac{2}{x}) + lim_{x \to 0} x + lim_{x \to 0} 1$

Step 2

Since $(e^{x} - x - 1) \cdot 0 \leq (e^{x} - x - 1) \cos^{2} (\frac{2}{x}) \leq (e^{x} - x - 1) \cdot 1$ and $lim_{x \to 0} (e^{x} - x - 1) \cdot 0 = lim_{x \to 0} (e^{x} - x - 1) \cdot 1 = 0$ , apply the squeeze theorem.

$0 + lim_{x \to 0} x + lim_{x \to 0} 1$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$0 + lim_{x \to 0} x + 1$

Step 4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$0 + 0 + 1$

Step 5

Simplify the answer.

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Step 5.1

Add $0$ and $0$ .

$0 + 1$

Step 5.2

Add $0$ and $1$ .

$1$