lim

x->0 sinx secx / x

use the fact that the limit as x approaches 1 of (sinx / x) = 1

Well, I'd like to recount a little joke while we figure out this math problem!

Why don't scientists trust atoms?
Because they make up everything!

Now, let's get back to the math. We can rewrite the given expression as (sinx * secx) / x. Since we're given that the limit as x approaches 0 of (sinx / x) equals 1, we can utilize this fact.

Using that fact, we can rewrite the expression as sinx * (1 / cosx) / x. By multiplying the numerator and denominator by cosx, we get sinx * (1 / cosx) * (1 / cosx) / (x * cosx). Simplifying this further, we have (sinx / cosx^2) * (1 / x * cosx).

Now, as x approaches 0, sinx / cosx approaches 0/1 = 0, and 1 / (x * cosx) approaches 1 / (0 * 1) = undefined. Therefore, the overall expression evaluates to 0 * undefined, which is undefined.

So, the limit of sinx * secx / x as x approaches 0 is undefined. Talk about a real math mystery!

To simplify the given limit, we'll start by multiplying the expression by the reciprocal of sec(x):

lim(x->0) [(sinx secx) / x] * [1 / secx]

This simplifies to:

lim(x->0) [sinx / (x * secx)]

Since sec(x) is the reciprocal of cos(x), we can rewrite the expression as:

lim(x->0) [sinx / (x * 1/cosx)]

Next, we'll multiply the expression by cos(x) / cos(x):

lim(x->0) [sinx / (x * 1/cosx)] * [cosx / cosx]

This simplifies to:

lim(x->0) [(sinx * cosx) / x]

Now, we can use the fact that lim(x->0) (sinx / x) = 1, to rewrite the expression as:

lim(x->0) [(sinx * cosx) / x] * [1 / cosx]

This further simplifies to:

lim(x->0) (sinx * cosx) / x * 1 / cosx

Finally, canceling out the common terms, we get:

lim(x->0) sinx / x

Using the given fact that lim(x->0) (sinx / x) = 1, we can conclude that:

lim(x->0) (sinx * secx) / x = 1

To solve the given limit, you can rewrite the expression using the limit identity that states:

lim(x->0) sinx/x = 1

Let's start by manipulating the given expression:

lim(x->0) sinx secx/x

Next, split the expression by multiplying and dividing it by secx:

lim(x->0) sinx (1/cosx) / x

Now, let's rewrite the expression in terms of sinx/x:

lim(x->0) sinx / x * 1/cosx

Since we know that lim(x->0) sinx/x = 1, we can substitute this value into our expression:

1 * 1/cosx

Simplifying the expression further:

1/cosx = secx

Therefore, the limit:

lim(x->0) sinx secx / x = 1 * sec(0) = 1 * sec(0) = 1 * 1 = 1

So, the value of the limit as x approaches 0 is 1.