I have been trying to figure out these limit problems and I get stuck on trying to move the function around to make it possible to solve.

The first one is:

lim sinx - cosx / tanx - 1
x--> (pi/4)

All I've gotten so far is to replace tanx with sinx/cosx but I don't even know if that's the way to go...

the second is:

lim sqrt(a + 2h) - sqrt(a) / h
h-->0

I tried multiplying by the conjugate, but when I multiply things out, I get stuck at:

2h / h*sqrt(a+2h) + sqrt(a)

pleaseeee help me. I have no idea how to move on from here; I'm terrible at figuring out what to do next in these situations.

For the first limit problem:

To simplify the expression, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (tanx + 1). This will help us eliminate the fraction and simplify the expression.

Let's rewrite the limit expression using the conjugate:

lim [sinx - cosx] / [tanx - 1] * [(tanx + 1) / (tanx + 1)]
x--> (pi/4)

Now, we can simplify the expression:

lim [sinx - cosx] * (tanx + 1) / [tanx - 1]
x--> (pi/4)

Next, expand and simplify the numerator:

lim [(sinx * tanx) + sinx - (cosx * tanx) - cosx] / [tanx - 1]
x--> (pi/4)

Now, using the identity tanx = sinx / cosx, we can replace the terms:

lim [(sinx * sinx / cosx) + sinx - (cosx * sinx / cosx) - cosx] / [tanx - 1]
x--> (pi/4)

Simplifying further, we get:

lim [sin^2(x) + sinx - cosx * sinx - cosx] / [tanx - 1]
x--> (pi/4)

Now, factor out sinx from the numerator:

lim [sinx(sinx + 1) - cosx(sinx + 1)] / [tanx - 1]
x--> (pi/4)

Factor out (sinx + 1) from both terms in the numerator:

lim [(sinx + 1)(sinx - cosx)] / [tanx - 1]
x--> (pi/4)

Now, simplify further:

lim (sinx - cosx) / (tanx - 1)
x--> (pi/4)

Since we have eliminated the indeterminate form of 0/0, we can directly substitute x = pi/4 into the expression:

(sin(pi/4) - cos(pi/4)) / (tan(pi/4) - 1)

Further simplifying, we get:

(1/sqrt(2) - 1/sqrt(2)) / (1 - 1)

Which simplifies to:

0 / 0

At this point, we have an indeterminate form. To evaluate this limit completely, we can use L'Hopital's Rule or another method. However, L'Hopital's Rule is not necessary in this case since we can observe that the numerator is 0 and the denominator is also 0. Therefore, the limit is indeterminate, meaning it does not have a finite value.

For the second limit problem:

Let's start from where you left off:

lim (2h) / (h * sqrt(a+2h) + sqrt(a))
h-->0

To simplify this expression, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (h * sqrt(a+2h) - sqrt(a)):

lim [(2h) * (h * sqrt(a+2h) - sqrt(a))] / [(h * sqrt(a+2h) + sqrt(a)) * (h * sqrt(a+2h) - sqrt(a))]
h-->0

Expanding the numerator:

lim [(2h^2) * sqrt(a+2h) - 2h * sqrt(a)] / [(h * sqrt(a+2h))^2 - (sqrt(a))^2]
h-->0

Simplifying the denominator:

lim [(2h^2) * sqrt(a+2h) - 2h * sqrt(a)] / [h^2 * (a+2h) - a]
h-->0

Next, we can cancel out the common factor of h in the numerator and denominator:

lim [2 * sqrt(a+2h) - 2 * sqrt(a)] / [h * (a+2h) - a]
h-->0

Now, let's substitute h = 0 into the expression:

[2 * sqrt(a+2(0)) - 2 * sqrt(a)] / [0 * (a+2(0)) - a]

Simplifying further, we get:

[2 * sqrt(a) - 2 * sqrt(a)] / -a

Which simplifies to:

(0) / -a = 0

Therefore, the limit of the expression as h approaches 0 is 0.

I hope this clears up any confusion and helps you understand how to approach these types of limit problems.

For the first limit problem, let's rearrange the expression in a way that makes it easier to solve.

lim (sinx - cosx) / (tanx - 1)
x --> (pi/4)

To proceed, recall the trigonometric identities:

sinx = cos(pi/2 - x)
tanx = sinx / cosx

Now, we can rewrite the expression as:

lim (cos(pi/2 - x) - cosx) / (sinx / cosx - 1)
x --> (pi/4)

Next, we can use the fact that cos(pi/2 - x) = sinx:

lim (sinx - cosx) / (sinx / cosx - 1)
x --> (pi/4)

To simplify further, let's multiply the numerator and denominator by cosx:

lim (sinx - cosx) * cosx / (sinx - cosx)
x --> (pi/4)

Now, cancel out the (sinx - cosx) terms:

lim cosx
x --> (pi/4)

Since cos(pi/4) = sqrt(2)/2, the limit is:

lim cosx = sqrt(2)/2
x --> (pi/4)

Moving on to the second limit problem:

lim sqrt(a + 2h) - sqrt(a) / h
h --> 0

To proceed, let's multiply the numerator and denominator by the conjugate of the numerator, which is sqrt(a + 2h) + sqrt(a):

lim [(sqrt(a + 2h) - sqrt(a)) * (sqrt(a + 2h) + sqrt(a))] / (h * (sqrt(a + 2h) + sqrt(a)))
h --> 0

Expanding the numerator gives:

lim (a + 2h - a) / (h * (sqrt(a + 2h) + sqrt(a)))
h --> 0

Simplifying further:

lim (2h) / (h * (sqrt(a + 2h) + sqrt(a)))
h --> 0

Canceling out the h terms:

lim 2 / (sqrt(a + 2h) + sqrt(a))
h --> 0

As h approaches zero, the term sqrt(a + 2h) approaches sqrt(a). Therefore, the limit becomes:

lim 2 / (2 * sqrt(a))
h --> 0

Canceling out the 2 terms:

lim 1 / sqrt(a)
h --> 0

The square root of a is a positive constant, so the limit simplifies to:

lim 1 / sqrt(a) = 1 / sqrt(a)
h --> 0

Therefore, the second limit is 1 / sqrt(a).