How would you find the limit of (secx-1)/(x^2) as x goes to 0 algebraically?

change secx -1 to 1/cosx -1 to (1-cosx)/cosx

then

lim (1-cosx)/(cosx)x^2 rationalize the numberator.

lim (1-cosx)(1+cosx)/(1+cosx)cosx x^2

lim (sin^2x)/x^2 * 1/(1+cosx)cosx

lim (sinx/x)lim sinx/x Lim (1/(cosx)(1+cosx)

1*1*1/(1*2)= 1/2

Multiply top and bottom by (sec(x)+1) to get

(secx-1)(sec(x)+1)/((x^2)(sec(x)+1)
=(sec²(x)-1)/((x^2)(sec(x)+1)
=(sin(x)/x)²/(cos²(x)(1+sec(x)))
Lim x->0 (sin(x)/x)=1
Lim x->0 cos(x)=1
Lim x->0 sec(x)=1
Therefore
Lim x->0 (secx-1)/(x^2) = 1/2

Alternatively, use l'Hôpital's rule.

To find the limit of (secx-1)/(x^2) as x approaches 0 algebraically, you can use L'Hôpital's rule. This rule states that if you have a limit of the form 0/0 or ∞/∞, you can take the derivative of the numerator and denominator separately and then evaluate the limit again.

Let's apply L'Hôpital's rule to find the limit:

1. Take the derivative of the numerator and denominator separately:

The derivative of secx is secx * tanx, and the derivative of x^2 is 2x.

So, the expression becomes (secx * tanx)/(2x).

2. Simplify the expression:

Distribute secx and cancel out an x from the numerator and denominator:

(secx * tanx)/(2x) = (tanx)/(2) = (sinx/cosx)/(2) = sinx/(2cosx).

3. Evaluate the limit as x approaches 0:

Plugging in x = 0, we get sin0/(2cos0) = 0/2 = 0.

Therefore, the limit of (secx-1)/(x^2) as x approaches 0 algebraically is 0.

To find the limit of (secx-1)/(x^2) as x approaches 0 algebraically, we can use a few algebraic manipulation techniques:

Step 1: Recall the identity secx = 1/cosx.

Step 2: Substitute 1/cosx for secx in the expression. Our expression now becomes (1/cosx - 1)/(x^2).

Step 3: Combine the fractions by getting a common denominator. Our expression can be rewritten as (1 - cosx)/(x^2 * cosx).

Step 4: Using the identity cosx = 1 - (1/2)x^2 + (1/24)x^4 - ..., we can rewrite cosx as 1 - (1/2)x^2 + higher order terms.

Step 5: Substituting the expression for cosx in our limit, we get [(1 - (1/2)x^2) - 1]/(x^2 * (1 - (1/2)x^2 + higher order terms)).

Step 6: Simplify the numerator. 1 - 1 = 0, so the numerator becomes -x^2/2.

Step 7: Simplify the denominator by multiplying out the terms and keeping only the highest degree of x. We get x^2 - (1/2)x^4 + higher order terms.

Step 8: Cancel out the common factor of x^2 in the numerator and denominator. Our expression now becomes -1/2 divided by (1 - (1/2)x^2 + higher order terms).

Step 9: Take the limit as x approaches 0. Since the higher order terms involve powers of x, as x approaches 0, those terms become negligible, leaving us with -1/2 divided by 1.

Step 10: Simplify the expression -1/2 divided by 1. This gives us -1/2.

Therefore, the limit of (secx-1)/(x^2) as x approaches 0 algebraically is -1/2.