Math  /  Calculus

Question29. limx0x2+93x2\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+9}-3}{x^{2}}
30. limx0x2+42x\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+4}-2}{x}
31. limx011x2x2\lim _{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x^{2}}
32. limx02x22x\lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2 x}

Studdy Solution
Evaluate the limit as x0 x \rightarrow 0 :
limx01x2+9+3=13+3=16 \lim_{x \rightarrow 0} \frac{1}{\sqrt{x^{2}+9}+3} = \frac{1}{3+3} = \frac{1}{6}
The value of the limit is:
16 \boxed{\frac{1}{6}}
### Problem 30: limx0x2+42x\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+4}-2}{x}
_ASSUMPTIONS_:
1. The expression involves a square root, suggesting rationalization.
2. Direct substitution of x=0 x = 0 results in an indeterminate form.

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