Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→5 1/x^2=1/25

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Start by recalling the precise definition of a limit: For every $ \varepsilon > 0 $, there exists a $ \delta > 0 $ such that if $ 0 < |x - 5| < \delta $, then $ \left| \frac{1}{x^2} - \frac{1}{25} \right| < \varepsilon $.

Simplify the expression $ \left| \frac{1}{x^2} - \frac{1}{25} \right| $ to find a relationship between $ \varepsilon $ and $ \delta $. This can be rewritten as $ \left| \frac{25 - x^2}{25x^2} \right| $.

Factor the numerator $ 25 - x^2 $ as $ (5 - x)(5 + x) $, so the expression becomes $ \left| \frac{(5 - x)(5 + x)}{25x^2} \right| $.

To ensure $ \left| \frac{(5 - x)(5 + x)}{25x^2} \right| < \varepsilon $, we need to bound $ |5 + x| $ and $ |x^2| $ by choosing $ \delta $ such that $ |x - 5| < \delta $ implies $ |x - 5| < 1 $, which gives $ 4 < x < 6 $.

Within this interval, $ |5 + x| $ is bounded by 11, and $ x^2 $ is bounded below by 16, so choose $ \delta $ such that $ \delta = \min\left(1, \frac{16\varepsilon}{11}\right) $ to satisfy the condition $ \left| \frac{(5 - x)(5 + x)}{25x^2} \right| < \varepsilon $.

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Limit Definition

The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value a, for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This formalism is essential for rigorously proving limits in calculus.

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In the context of limits, the ε (epsilon) represents how close f(x) must be to the limit L, while δ (delta) represents how close x must be to the point a. Establishing a relationship between ε and δ is crucial for demonstrating that as x gets sufficiently close to a, f(x) will be within ε of L, thus proving the limit exists.

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A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. In this case, since 1/x^2 is continuous around x = 5, we can directly apply the limit definition without concerns about discontinuities, simplifying the proof process.

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