Simplifying and evaluating the limit of (x² + x) / sin(x) as x approaches 0

Evaluate the limit Lim (x²+x)/(sin(x))x→0

To evaluate the given limit:

lim (x² + x) / sin(x) as x approaches 0,

we can try to simplify the expression in order to apply limit properties

To evaluate the given limit:

lim (x² + x) / sin(x) as x approaches 0,

we can try to simplify the expression in order to apply limit properties.

First, let’s factor out x from the numerator:

lim x(x + 1) / sin(x) as x approaches 0.

Next, let’s consider the limit of the numerator and denominator separately:

lim x → 0 (x) * lim x → 0 (x + 1) / lim x → 0 (sin(x)).

The limit of x as x approaches 0 is simply 0:

0 * lim x → 0 (x + 1) / lim x → 0 (sin(x)).

Now let’s evaluate the limit of x + 1 as x approaches 0:

0 * (0 + 1) / lim x → 0 (sin(x)).

This simplifies to:

0 / lim x → 0 (sin(x)).

Finally, let’s evaluate the limit of sin(x) as x approaches 0:

0 / (sin(0)).

Since sin(0) is equal to 0, we have:

0 / 0.

However, this is an indeterminate form and does not provide a definite value for the limit.

In this case, we can use L’Hopital’s rule to find the limit.

Taking the derivative of the numerator and denominator and applying L’Hopital’s rule:

lim x → 0 (1 + 0) / cos(x).

Simplifying further:

lim x → 0 (1 / cos(x)).

Now, as x approaches 0, the cosine function approaches 1:

1 / cos(0) = 1 / 1 = 1.

Therefore, the limit of (x² + x) / sin(x) as x approaches 0 is equal to 1.

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