Question

webwork 5 - topics 10 - 12: pro
(1 point)
find the following limit using lhopitals rule:

$$ lim _ { x ightarrow 0 ^ { + } } \frac { int _ { 0 } ^ { x } sqrt { t } cos t d t } { x ^ { 2 } } = $$

help (limits)

Explanation:

Step1: Check indeterminate form

Substitute $x=0^+$: numerator $\int_0^0 \sqrt{t}\cos t dt = 0$, denominator $0^2=0$. So $0/0$.

Step2: Apply L'Hospital's Rule

Differentiate numerator (Fundamental Theorem of Calculus) and denominator: $\frac{\sqrt{x}\cos x}{2x}$.

Step3: Simplify expression

$\frac{\sqrt{x}\cos x}{2x} = \frac{\cos x}{2\sqrt{x}}$.

Step4: Evaluate limit as $x\to0^+$

$\cos 0 = 1$, so $\frac{1}{2\sqrt{x}} \to +\infty$.

Answer:

$+\infty$