Exploring Limits and Integrals in High School Mathematics

A) Finding the Limit: lim (x→∞) arctan(e^x)

When dealing with limits, it's essential to understand the behavior of the function as the input variable approaches a specific value. In this case, we're interested in the limit of the function arctan(e^x) as x approaches positive infinity.

The arctan (or inverse tangent) function is a trigonometric function that maps the range of real numbers to the interval (-π/2, π/2). The exponential function e^x grows without bound as x approaches positive infinity.

To find the limit of arctan(e^x) as x approaches positive infinity, we can use the properties of the arctan function and the behavior of the exponential function.

As x approaches positive infinity, the value of e^x also approaches positive infinity. This means that the input to the arctan function, e^x, will become increasingly large in magnitude.

The arctan function has the property that as its input approaches positive or negative infinity, the function value approaches π/2 or -π/2, respectively.

Therefore, the limit of arctan(e^x) as x approaches positive infinity is:

lim (x→∞) arctan(e^x) = π/2

This means that the function arctan(e^x) approaches the value π/2 as x approaches positive infinity.

B) Evaluating the Integral: ∫₀^(√3/5) dx/(1 + 25x^2)

Evaluating integrals is a fundamental skill in calculus and high school mathematics. In this case, we're asked to evaluate the integral from 0 to (√3)/5 of the function dx/(1 + 25x^2).

To evaluate this integral, we can use the technique of substitution. Let's define a new variable u = 1 + 25x^2, so du = 50x dx.

Substituting, we get:

∫₀^(√3/5) dx/(1 + 25x^2) = (1/50) ∫₁^(1 + 25(√3/5)^2) du/u

Evaluating the integral, we have:

(1/50) ∫₁^(1 + 25(√3/5)^2) du/u = (1/50) [ln(u)]₁^(1 + 25(√3/5)^2)

Substituting the limits, we get:

(1/50) [ln(1 + 25(√3/5)^2) - ln(1)] = (1/50) ln(1 + 25(√3/5)^2)

Therefore, the value of the integral from 0 to (√3)/5 of dx/(1 + 25x^2) is:

(1/50) ln(1 + 25(√3/5)^2)

This expression provides the exact value of the integral within the given limits.