Question

question 10 find the limit: \\(\lim_{x \to 196} \frac{\sqrt{x} - 14}{x - 196}\\) give an exact answer or keep at least three decimal places. question help: \\(\square\\) video

Explanation:

Step1: Recognize indeterminate form

Substitute $x=196$: $\frac{\sqrt{196}-14}{196-196}=\frac{14-14}{0}=\frac{0}{0}$

Step2: Rationalize the numerator

Multiply numerator/denominator by $\sqrt{x}+14$:
$$\lim_{x \to 196} \frac{(\sqrt{x}-14)(\sqrt{x}+14)}{(x-196)(\sqrt{x}+14)}$$

Step3: Simplify the expression

Use $(a-b)(a+b)=a^2-b^2$:
$$\lim_{x \to 196} \frac{x-196}{(x-196)(\sqrt{x}+14)} = \lim_{x \to 196} \frac{1}{\sqrt{x}+14}$$

Step4: Substitute $x=196$

$$\frac{1}{\sqrt{196}+14} = \frac{1}{14+14}$$

Step5: Compute final value

$$\frac{1}{28} \approx 0.0357$$

Answer:

$\frac{1}{28}$ or $0.036$ (rounded to three decimal places)