Why does e^x approach infinity as x approaches infinity in mathematics? Understanding the exponential function

e^x när x->∞

When x approaches infinity, e^x also approaches infinity

When x approaches infinity, e^x also approaches infinity.

To understand why, let’s recall the definition of the exponential function e^x. The exponential function e^x is defined as the sum of the infinite series:

e^x = 1 + x/1! + x^2/2! + x^3/3! + …

As x increases, each term in the series grows larger and larger. The factorial terms in the denominator (e.g., 1!, 2!, 3!) increase at a slower rate compared to the corresponding power of x in the numerator (e.g., x^1, x^2, x^3). Consequently, the terms in the series eventually dominate the denominator, causing the sum to rise exponentially.

In simpler terms, as x becomes infinitely large, the exponential function e^x grows without bound. This is primarily due to the fact that the exponential function is defined as the limit of the exponential series as the number of terms approaches infinity.

Therefore, when x approaches infinity, e^x also approaches infinity.

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