Question

Indeterminate of the form 1 [infinity] Indeterminate of the form [infinity] 0 Indeterminate of the form 00 ​ Indeterminate of the form [infinity][infinity] ​ Part 2 of 7 The limit is indeterminate of the form 1 [infinity] . Write the function f(x)=(1+ x^7​ ) 3x as an exponential function. lim x→[infinity]​ (1+ x^7​ ) 3x =lim x→[infinity]​ x

Answer 1

The given limit lim (x→∞)
`(1 + x^7)^(3x)` is an indeterminate form of 1^∞. To simplify this expression, we can rewrite the function as an exponential function. The answer is lim (x→∞)
`(1 + x^7)^(3x)` =
`e^{\lim` (x→∞)
`3x * ln(1 + x^7)} = e^(\infty)` = ∞.

Step-by-step explanation:

In the given problem, we have the limit lim (x→∞)
`(1 + x^7)^(3x)` , which is in an indeterminate form of 1^∞. To simplify this expression, we can use the properties of exponential functions.

Let
`f(x) = (1 + x^7)^(3x)`. Taking the natural logarithm of both sides, we get
`ln(f(x)) = 3x * ln(1 + x^7).`

Now, we can rewrite the original limit as

lim (x→∞)
`(1 + x^7)^(3x)` =
`e^{\lim` (x→∞)
`3x * ln(1 + x^7)} = e^(\infty)` = ∞.

Next, we focus on the expression within the exponent, which is
`3x * ln(1 + x^7)`.

As x approaches infinity, the term
`x^7` becomes dominant, and
`ln(1 + x^7)` approaches infinity. Therefore, the limit becomes 3x * ∞, resulting in an indeterminate form of ∞ * ∞. To resolve this, we can rewrite the limit as e^
`{\infty}`, which simplifies to ∞.

In summary, the limit lim (x→∞)
`(1 + x^7)^(3x)` is an indeterminate form of 1^∞, and by expressing it as an exponential function and applying the properties of logarithms, we find that the final answer is e^∞, which evaluates to ∞.