Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.

lim x→∞ x^7e^−x^6

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim x→∞ (x tan(3/x))

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim x→1 ((7x/(x−1)-(7/(ln x))

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim x→0+ (tan 5x)^x

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim x→∞ x^(4/x)

To find the limits given, we can determine if l'Hospital's Rule is applicable. If the limit is in an indeterminate form of the type 0/0 or ∞/∞, we can apply l'Hospital's Rule. If not, we need to use a different method.

1. lim x→∞ x^7e^−x^6:
The limit is in the form ∞⋅0, which is an indeterminate form where l'Hospital's Rule can be applied. To use l'Hospital's Rule, we take the derivative of the numerator and denominator separately, and then find the limit again.

Differentiating the numerator: d/dx (x^7) = 7x^6
Differentiating the denominator: d/dx (e^−x^6) = -6x^5 * e^−x^6

Now, we can find the limit again:
lim x→∞ (7x^6)/(-6x^5 * e^−x^6)

The limit has become -∞/∞, which is still an indeterminate form. We can apply l'Hospital's Rule one more time. Differentiate the numerator and denominator again:

Differentiating the numerator: d/dx (7x^6) = 42x^5
Differentiating the denominator: d/dx (-6x^5 * e^−x^6) = -30x^4 * e^−x^6 - (6x^5 * -6x^5 * e^−x^6)

Now, we find the limit once more:
lim x→∞ (42x^5)/(-30x^4 * e^−x^6 - (6x^5 * -6x^5 * e^−x^6))

If we continue this process, we will eventually have a limit that is not in an indeterminate form, and we can determine the overall limit.

2. lim x→∞ (x tan(3/x)):
The limit is also in an indeterminate form of ∞⋅0. To apply l'Hospital's Rule, we differentiate both numerator and denominator:

Differentiating the numerator: d/dx (x) = 1
Differentiating the denominator: d/dx (tan(3/x)) = sec^2(3/x) * (-3/x^2)

Now, we find the limit again:
lim x→∞ (1)/(sec^2(3/x) * (-3/x^2))

This limit is still in an indeterminate form. We can apply l'Hospital's Rule once more by differentiating the numerator and denominator.

Differentiating the numerator: d/dx (1) = 0
Differentiating the denominator: d/dx (sec^2(3/x) * (-3/x^2)) = -6 * sec^2(3/x) * (tan(3/x))/x^3

Now, we find the limit again:
lim x→∞ (0)/(-6 * sec^2(3/x) * (tan(3/x))/x^3)

After one more application of l'Hospital's Rule, the limit should converge to a non-indeterminate form.

3. lim x→1 ((7x/(x−1)-(7/(ln x))):
The limit is not in an indeterminate form, so we need to use an alternative method.

In this case, we can try simplifying the expression to make it easier to evaluate the limit. Expanding the numerator, we get:

((7x/(x−1))- (7/(ln x)))
= ((7x^2 - 7)/(x(x-1))) - (7/(ln x))

Now, we can simplify further:
= (7(x^2 - 1))/(x(x-1)) - (7/(ln x))

Next, we factor out 7:
= (7(x+1)(x-1))/(x(x-1)) - (7/(ln x))

Now, we can cancel out (x-1) terms:
= (7(x+1))/x - (7/(ln x))

Finally, we can evaluate the limit:
lim x→1 (7(x+1))/x - (7/(ln x))
= (7(1+1))/1 - (7/(ln 1))
= 14 - 0
= 14

So, the limit is 14.

4. lim x→0+ (tan 5x)^x:
The limit is not in an indeterminate form, so we need to use an alternative method.

In this case, we can try taking the natural logarithm of the expression to simplify it. We have:

ln(lim x→0+ (tan 5x)^x)

Now, we can use the property of logarithms to bring the exponent down:
lim x→0+ x*ln(tan 5x)

Next, we can apply the limit property of logarithms:
lim x→0+ ln(tan 5x) / (1/x)

Now, we simplify further by applying the limit property of natural logarithm:
ln(lim x→0+ tan 5x) / (lim x→0+ 1/x)

The limit of tan 5x as x approaches 0 is 0, and the limit of 1/x as x approaches 0 is ∞. Therefore, we have:

ln(0) / ∞

This is an indeterminate form. We can use l'Hospital's Rule to differentiate the numerator and denominator:

Differentiating the numerator: d/dx (ln(0)) = 0
Differentiating the denominator: d/dx (∞) = ∞

Now, we can find the limit:
lim x→0+ 0 / ∞

We can simplify this limit as 0. Therefore, the final limit is:

0

5. lim x→∞ x^(4/x):
The limit is not in an indeterminate form, so we need to use an alternative method.

In this case, we can try taking the natural logarithm of the expression to simplify it. We have:

ln(lim x→∞ x^(4/x))

Now, we can use logarithmic properties to bring down the exponent:
lim x→∞ (4/x) * ln(x)

Next, we simplify further by applying the limit property of natural logarithm:
(4/∞) * ln(∞)

The limit of ln(∞) is ∞, and multiplying by 4/∞ gives us a final limit of:

0

So, the limit is 0.